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JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA PDF Download

Q.1. A vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA lies in the plane of the vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA If JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA bisects the angle between JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then    (2020)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(3)
We have,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
On comparing with JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
We get α = 4 and β = 4
Therefore, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now, again consider JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
On comparing withJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA we get
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Therefore, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.2. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be three unit vectors such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then the ordered pair, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to    (2020)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(4)
Given
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(1)
Taking square of both sides, we get
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA (since JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are unit vectors)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.3. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be two vectors. If JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is a vector such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to    (2020)
(1) -3/2
(2) 1/2
(3) -1/2
(4) -1
Ans.
(3)
Given JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(1)
Now, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Substituting these values in Eq. (1), we get
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.4. The projection of the line segment joining the points (1, −1, 3) and (2, −4, 11) on the line joining the points (−1, 2, 3) and (3, −2, 10) is _____.    (2020)
Ans.
(8.00)
We have
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, the required projection is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.5. If the vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are coplanar and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then the value of λ is _____.    (2020)
Ans.
(1.00)
The vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are coplanar vectors, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Therefore, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(2)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(3)
Now, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(4)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(5)
Hence, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.6. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be three vectors such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and the angle between JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA If JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular to the vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to ____.    (2020)
Ans.
(30.00)
Given, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular to the vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.7. If the distance between the plane 23x - 10y - 2z + 48 =0 and the plane containing the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then k is equal to _____.    (2020)
Ans.
(3.00)
The given lines must be intersecting

Therefore,
(2s - 1, 3s + 3, 8s - 1) = (2t - 3t, t - 2, λt + 1)
Now, 2s - 1 = 2t - 3 ...(1)
3s + 3 = t - 2 ...(2)
8s - 1 = λt + 1 ...(3)
On solving above equations, we get  
t = -1, s = -2 and λ = 18
Distance of plane contains given lines from given plane is same as distance between point (–3, –2, 1) from given plane. So,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.8. Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then, the image of R in the plane P is    (2020)
(1) (6, 5, 2)
(2) (6, 5, −2)
(3) (4, 3, 2)
(4) (3, 4, −2)
Ans.
(2)
The equation of the plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ (x - 2)(1) - (y - 1)(2 - 3) + z(-2) = 0
⇒ x + y - 2z = 3
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Let I and F are respectively image and foot of perpendicular of point R in the plane. So, the equation of RI is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA (say)
⇒ x = k + 2, y = k + 1 and z = -2k + 6
Hence, the coordinates of point F is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA 
Point F lies in the plane, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ 3k = 12 ⇒ k = 4
Hence, the image I is (6, 5, -2).

Q.9. If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (α, 7, 1) is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA  then α is equal to _____.
Ans.
(4.00)
We have,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Direction ratio of BP JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Direction ratio of AP =JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Since AP and BP are perpendicular, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.10. The shortest distance between the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is    (2020)
(1) 2√30
(2) 7/2 (√30)
(3) 3√30
(4) 3
Ans.
(3)
The equations of the lines are
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA  ...(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(2)
The distance between the lines is given by
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.11. Let the volume of a parallelepiped whose coterminous edges are given by JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be 1 cu. unit. If θ be the angle between the edges JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then cos θ can be    (2020)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) 5/7
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(2)
The volume of parallelepiped is 1 cu. unit. Therefore,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ λ = 2, 4
Now, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.12. The mirror image of the point (1, 2, 3) in a plane is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA Which of the following points lies on this plane?    (2020)
(1) (1, 1, 1)
(2) (1, -1, 1)
(3) (-1, -1, 1)
(4) (-1, -1, -1)
Ans.
(2)
Direction ratios of normal to the plane is
<1 + (7/3), 2 + (4/3),3 + (1/3)> = <1, 1, 1>
The coordinates of mid- point of image and point is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
The equation of plane is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ x + y + z = 1
Hence, the point (1, −1, 1) lies on the plane.

Q.13. If for some α and β in JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAthe intersection of the following three planes
x + 4y - 2z =1
x + 7y - 5z = β
x + 5y - αz = 5
is a line in JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAthen α + β is equal to    (2020)
(1) 0
(2) 10
(3) 2
(4) −10
Ans.
(2)
We have
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA=0
⇒ (7α + 25) - (4α + 10) + (-20 + 14) = 0
⇒ 3α + 9 = 0 ⇒ α = -3
and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, α + β = -3 + 13 = 10

Q.14. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be a vector such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to:    (2019)
(1) 19/2
(2) 9
(3) 8
(4) 17/2
Ans.
(1)
 JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.15. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be three vectors such that the projection vector of JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
If JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to:    (2019)
(1) √32
(2) 6
(3) √22
(4) 4
Ans.
(2)
Projection of JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
According to question JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
 b1 + b2 = 2    ...(1)
Since, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular toJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ 8 + 5b1 + b2 + 2 = 0 ...(2)
From (1) and (2),
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.16. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be three vectors such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA Then a possible value of (λ1, λ2, λ3) is:    (2019)
(1) (1, 3, 1)
(2) (-1/2, 4, 0)
(3) (1/2, 4, -2)
(4) (1, 5, 1)
Ans.
(2)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ 3 - λ2 = 2λ1    ...(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(2)
Since, (-1/2, 4, 0) satisfies equation (1) and (2). Hence, one of possible value of
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.17. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be two given vectors where vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAare non-collinear. The value of λ for which vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are collinear, is:    (2019)
(1) -4
(2) -3
(3) 4
(4) 3
Ans.
(1)
Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are collinear for same k
i.e., JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
But JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are non-collinear, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
λ = -4

Q.18. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAand JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be coplanar vectors. Then the non-zero vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans. 
(4)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are coplanar
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
For JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA(Rejected)

Q.19. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3/√2, then the sum of all possible values of β is:    (2019)
(1) 4 
(2) 3
(3) 2
(4) 1
Ans.
(4)
Since, the angle bisector of acute angle between OA and OB would be y = x
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Since, the distance of C from bisector = 3/√2
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, the sum of all possible value of β = 2 + (-1) = 1

Q.20. The sum of the distinct real values of μ, for which the vectors, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are co-planar, is:    (2019)
(1) -1
(2) 0
(3) 1
(4) 2
Ans.
(1)
∵ Three vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAand JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are copalnar.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Therefore, sum of all real values = 1 - 2 = -1

Q.21. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be three unit vectors, out of which vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAare non-parallel. If α and β are the angles which vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA makes with vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA respectively and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to:    (2019)
(1) 30°
(2) 90°
(3) 60°
(4) 45°
Ans.
(1)
Since, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are three unit vectors
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Then, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ β = 60° and α = 90°
Hence, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.22. The magnitude of the projection of the vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA on the vector perpendicular to the plane containing the vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2019)
(1) √3/2
(2) √6
(3) 3√6
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans. 
(4)
Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ Vector perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now, projection of vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.23. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA for some real x. Then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is possible if:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(2)
Given JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.24. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAwhere JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is parallel to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(3)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(1)
Since, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Cross product with JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA in equation (1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.25. If a unit vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA makes angles π/3 with JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and θ ∈ (0, π) with JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then a value of 0 is:    (2019)
(1) 5π/6
(2) π/4
(3) 5π/12
(4) 2π/3
Ans.
(4)
Let cos α, cos β, cos γ be direction cosines of a.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.26. The distance of the point having position vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA from the straight line passing through the point (2, 3, -4) and parallel to the vector, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2019)
(1) 7
(2) 4√3
(3) 6
(4) 2√13
Ans. 
(1)
Equation of the line is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Let the point M on the line l is (6λ + 2, 3λ + 3, -4λ - 4)
Direction ratio's of PM is (6λ + 3, 3λ + 1, - 4λ- 10)
⇒ (6λ + 3) (6) + (3λ + 1) (3) + (-4λ.- 10) (- 4) = 0
⇒ λ = - 1 ⇒ M = (- 4, 0, 0)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.27. Let a = 3i + 2j + 2k and b = i + 2j - 2k be two vectors. If a vector perpendicular to both the vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA has the magnitude 12 then one such vector is:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(2)
Let vector be JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Given that magnitude of the vector is 12.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.28. Let α ∈ R and the three vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA Then the set JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA    (2019)
(1) is singleton
(2) is empty
(3) contains exactly two positive numbers
(4) contains exactly two numbers only one of which is positive
Ans.
(2)
Let, three vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are coplanar, then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA 
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∵ no real value of ‘α’ exist,
∴ set S is an empty set.

Q.29. The equation of the line passing through (-4, 3, 1), parallel to the plane x + 2y - z - 5 = 0 and intersecting the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(3)
Let any point on the intersecting line
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
is (-3λ - 1, 2λ + 3, -λ + 2)
Since, the above point lies on a line which passes through the point (-4, 3, 1)
Then, direction ratio of the required line
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Since, line is parallel to the plane
x + 2y - z - 5 = 0
Then, perpendicular vector to the line is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now (-3λ + 3)(1) + (2λ)(2) + (-λ + 1)(-1) = 0
⇒  λ = - 1
Now direction ratio of the required line = <6, - 2, 2 > or <3,-1, 1>
Hence required equation of the line is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.30. The plane through the intersection of the planes x + y + z = 1 and 2x + 3y-z + 4 = 0 and parallel to j-axis also passes through the point:    (2019)
(1) (-3, 0, -1)
(2) (-3, 1, 1)
(3) (3, 3, -1)
(4) (3, 2, 1)
Ans.
(4)
Since, equation of plane through intersection of planes
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
But, the above plane is parallel to y-axis then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ λ = -3
Hence, the equation of required plane is
-x - 4z + 7 = 0
⇒ x + 4z - 7 = 0
Therefore, (3, 2, 1) the passes through the point.

Q.31. The equation of the plane containing the straight line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and perpendicular to the plane containing the straight lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2019)
(1) x - 2y + z = 0
(2) 3x + 2y - 3z = 0
(3) x + 2y - 2z = 0
(4) 5x + 2y - 4z = 0
Ans.
(1)
Let the direction ratios of the plane containing lines
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ Direction ratio of plane = <-8, 1, 10>.
Let the direction ratio of required plane is <l, m, n>
Then-81 + m + 10n = 0    ...(1)
and 2l + 3m + 4n = 0    ...(2)
From (1) and (2),
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ D.R.s are <1, -2, 1>

∴ Equation of plane: x - 2y + z = 0

Q.32. The plane passing through the point (4, -1, 2) and parallel to the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA also passes through the point:    (2019)
(1) (1, 1, -1)
(2) (1, 1, 1)
(3) (-1, -1,-1)
(4) (-1, -1, 1)
Ans. 
(2)
Equation of required plane is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(x - 4)(-3 -4) - (y + 1)(9 - 2) + (z - 2) (6 + 1) = 0
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∵ point (1, 1, 1) satisfies this equation
∴ point (1, 1, 1) lies on the plane

Q.33. Let A be a point on the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and B(3, 2, 6) be a point in the space. Then the value of μ for which the vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is parallel to the plane x - 4y + 3z = 1 is:    (2019)
(1) 1/4
(2) 1/8
(3) 1/2
(4) -1/4
Ans. 
(1)
∵ A be a point on given line.
∴ Position vector of A
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Equation of plane is: x - 4y + 3z = 1
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is parallel to this plane.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ μ = 1/4

Q.34. The plane which bisects the line segment joining the points (- 3, - 3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?    (2019)
(1) (-2, 3, 5)
(2) (4, -1, 7)
(3) (2, 1, 3)
(4) (4, 1, -2)
Ans.
(4)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Q(0, 2, 5)
Since, direction ratios of normal to the plane is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Then, equation of the plane is
(x - 0)6 + (y - 2)10 + (z - 5)2 = 0
3x + 5y - 10 + z - 5 = 0
3x + 5y + z = 15    ...(1)
Since, plane (1) satisfies the point (4, 1, -2)
Hence, required point is (4, 1, -2)

Q.35. On which of the following lines lies the point of inter-section of the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAand the plane, x + y + z = 2?    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(3)
Let any point on the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA be A(2λ + 4, 2λ + 5, λ, + 3) which lies on the plane x +y + z = 2
 2λ + 4 + 2λ + 5 + λ + 3 = 2
 5λ = -10 ⇒ λ = - 2
Then, the point of intersection is (0, 1, 1)
which lies on the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.36. The plane containing the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and also containing its projection on the plane 2x + 3y- z = 5, contains which one of the following points?    (2019)
(1) (2, 2, 0) 
(2) (-2, 2, 2)
(3) (0, -2, 2)
(4) (2, 0, -2)
Ans.
(4)
Let normal to the required plane is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is perpendicular to both vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ Equation of the required plane is
 (x - 3)(-8) + (y + 2) x 8 + (z - 1) x 8 = 0
⇒ (x - 3) (-1) + (y + 2) x 1 + (z - 1) x 1 = 0
 x - 3 - y - 2 - z + 1 =0
∵ x - y - z = 4 passes through (2, 0, -2)
∴ plane contains (2, 0, -2).

Q.37. The direction ratios of normal to the plane through the points (0, -1, 0) and (0,0, 1) and making an angle π/4 with the plane y - z + 5 = 0 are:    (2019)
(1) 2, -1, 1
(2) 2, √2, -√2
(3) √2, 1, -1
(4) 2√3, 1, -1
Ans.
(2, 3)
Let the d.r’s of the normal be (a, b, c)
Equation of the plane is
a(x - 0) + b(y + 1) + c(z - 0) = 0
∵ It passes through (0, 0, 1)
∴ b + c
Also JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ The d.r’s are √2, 1, -1 or 2, √2, -√2

Q.38. Two lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA intersect at the point R. The reflection of R in the xy-plane has coordinates:    (2019)
(1) (2, -4, -7)
(2) (2, 4, 7)
(3) (2, -4, 7)
(4) (-2, 4, 7)
Ans.
(1)
Let the coordinate of P with respect to line
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
and coordinate of P w.r.t.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
From above equation : λ = -1, μ = 1 .
∴ Coordinate of point of intersection R = (2, -4, 1).
Image of R w.r.t. xy plane = (2, -4, -7).

Q.39. If the point (2, α, β) lies on the plane which passes through the points (3,4, 2) and (7, 0, 6) and is perpendicular to the plane 2x-5y = 15, then 2α - 3β is equal to:    (2019)
(1) 12
(2) 7
(3) 5
(4) 17
Ans.
(2)
Let the normal to the required plane is n, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ Equation of the plane
(x - 3) x 20 + (y - 4) x 8 + (z - 2) x (-12) = 0
5x - 15 + 2y - 8 - 3z + 6 = 0
5x + 2y - 3z - 17 = 0    ...(1)
Since, equation of plane (1) passes through (2, α, β), then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.40. The perpendicular distance from the origin to the plane containing the two lines, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2019)
(1) 11√6
(2) 11/√6
(3) 11
(4) 6/√11
Ans.
(2)
∵ plane containing both lines.
∴ D.R. of plane JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now, equation of plane is,
7(x - 1) - 14(y - 4) + 7 (z + 4) = 0
 x - 1 - 2y + 8 + z + 4 = 0
⇒ x - 2y + z + 11 = 0
Hence, distance from (0, 0, 0) to the plane,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.41. If an angle between the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and the plane, x - 2y - kx = 3 is  JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAthen a value of k is    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) -3/5
(4) -5/3
Ans.
(1)
Let angle between line and plane is θ, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Since, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Then, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.42. Let S be the set of all real values of λ such that a plane passing through the points {-λ2, 1, 1), (1, -λ2, 1) and (1, 1, -λ2) also passes through the point-(-1,-1, 1). Then S is equal to:    (2019)
(1) {√3}
(2) {√3, -√3}
(3) {1, -1}
(4) {3, -3}
Ans.
(2)
Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA D(-1, -1, 1) lie on same plane, then
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ (λ2 + 1)((1 -λ2)-4) = 0
 (3 - λ2)(λ+ 1) = 0 ⇒ λ2 = 3
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, S = {-√3, √3}

Q.43. The equation of a plane containing the line of intersection of the planes 2x - y - 4 = 0 and y + 2z - 4 = 0 and passing through the point (1, 1, 0) is:    (2019)
(1) x- 3y- 2z = -2
(2) 2x - z = 2
(3) x - y - z = 0
(4) x + 3y + z = 4
Ans.
(3)
Let the equation of required plane be;
(2x - y - 4) + λ(y + 2z - 4) = 0
∴  This plane passes through the point (1, 1, 0) then (2 - 1 - 4) + λ(1 + 0 - 4) = 0
⇒ λ = -1
Then, equation of required plane is,
(2x - y - 4) - (y + 2z - 4) = 0
⇒ 2x - 2y — 2z = 0 ⇒ x - y - z = 0

Q.44. The length of the perpendicular from the point (2, -1, 4) on the straight line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis:    (2019)
(1) greater than 3 but less than 4
(2) less than 2
(3) greater than 2 but less than 3
(4) greater than 4
Ans.
(1)
Let P be the foot of perpendicular from point T (2, -1, 4) on the given line. So P can be assumed as P(10λ - 3, -7λ + 2, λ)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
DR's of TP is proportional to 10λ - 5, - 7λ + 3, λ - 4
∵ TP and given line are perpendicular, so
10(10λ - 5) -7 (- 7λ + 3)+ 1(λ - 4) = 0
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, the length of perpendicular is greater than 3 but less than 4.

Q.45. The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0 is:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(4)
Equation of the plane passing through the line of intersection of x + y + z = 1 and 2x + 3y + 4z = 5 is (2x + 3y + 4z - 5) + λ (x + y + z - 1) = 0
∵ plane (i) is perpendicular to the plane x-y + z = 0
∴ (2 + λ)(1) + (3 + λ)(- 1) + (4 + λ)(1) = 0
2 + λ - 3 - λ + 4 + λ = 0 
⇒ λ = -3
Hence, equation of required plane is -x + z- 2 = 0 or x - z + 2 = 0
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.46. If a point R(4, y, z) lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10), then distance of R from the origin is:    (2019)
(1) 2√14
(2) 2√21
(3) 6
(4) √53
Ans.
(1)
Here, P, Q, R are collinear
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now, or = JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.47. If the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is:    (2019)
(1) √5 / 2
(2) 2 / √5
(3) 9/2
(4) 7/2
Ans.
(3)
Let point on line be P (2k + 1, 3k - 1, 4k + 2)
Since, point P lies on the plane x + 2y + 3z = 15
∴ 2k + 1 + 6k - 2 + 12k + 6= 15
⇒ k = 1/2
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Then the distance of the point P from the origin is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.48. A plane passing through the points (0, -1, 0) and (0, 0, 1) and making an angle π/4 with the plane y - z + 5 = 0, also passes through the point:    (2019)
(1) (-√2, 1, -4)
(2) (-√2, -1, 4)
(3) (-√2, -1, -4)
(4) (√2, 1, 4)
Ans.
(4)
Let the required plane passing through the points (0,-1, 0) and (0, 0, 1) be JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and the given plane is y - z + 5 = 0
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Then, the equation of plane is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Then the point (√2, 1, 4) satisfies the equation of plane -√2x - y + z = 1

Q.49. The vertices B and C of a ΔABC lie on the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA such that BC = 5 units. Then the area (in sq. units) of this triangle, given that the point A (1, -1, 2), is:    (2019)
(1) 5√17
(2) 2√34
(3) 6
(4) √34
Ans.
(4)
Let a point D on BC = (3λ - 2, 1, 4λ)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Area of triangle JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.50. Let P be the plane, which contains the line of intersection of the planes, x + y + z - 6 = 0 and 2x + 3y + z + 5 = 0 and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to:    (2019)
(1) 17/√5
(2) 63√5
(3) 205√5
(4) 11/√5
Ans.
(4)
Let the plane be
P = (2x + 3y + z + 5) + λ(x: +y + z- 6) = 0
∵ above plane is perpendicular to xy plane.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, the equation of the plane is,
P = x + 2y+ 11 = 0
Distance of the plane P from (0, 0, 256)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.51. If Q (0, -1, -3) is the image of the point P in the plane 3x - y + 4z = 2 and R is the point (3, -1, -2), then the area (in sq. units) of ΔPQR is:    (2019)
(1) 2√13
(2) √91/4
(3) √91/2
(4) √65/2
Ans.
(3)
Image of Q (0, -1, -3) in plane is,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.52. Let A(3, 0, -1), B(2, 10, 6) and C (1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then cos (∠GOA)(O being the origin) is equal to:    (2019)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) 1/√15
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) 1/√30
Ans.
(2)
G is the centroid of ΔABC.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.53. If the length of the perpendicular from the point (β, 0, β) (β ≠ 0) to the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then β is equal to:    (2019)
(1) 1
(2) 2
(3) -1
(4) -2
Ans.
(3)
Given, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA (let) and point P (β, 0, β)
Any point on line A = (p, 1, -p -1)
Now, DR of AP ≡ <p - β, 1 - 0 - p - 1 - β>
Which is perpendicular to line.
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Given that distance AP JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
∴ β = -1

Q.54. A perpendicular is drawn from a point on the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x - y + z = 3. Then the co-ordinates of Q are:    (2019)
(1) (1, 0, 2)
(2) (2, 0, 1)
(3) (-1, 0, 4)
(4) (4, 0, -1)
Ans. 
(2)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Let co-ordinates of Q be (α, β, γ), then
α + β + γ = 3    ...(i)
α - β + γ = 3    ...(ii)
⇒ α + γ = 3 and β = 0
Equating direction ratio's of PQ, we get
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Substituting the values of a and y in equation (i), we get
⇒ 5λ + 3 = 3 ⇒ λ = 0
Hence, point is Q (2, 0, 1)

Q.55. If the plane 2x - y + 2z + 3 =0 has the distances 1/2 and 2/3 units from the planes 4x - 2y + 4z + λ = 0 and 2x - y + 2z + μ = 0, respectively, then the maximum value of λ + μ is equal to:    (2019)
(1) 9
(2) 15
(3) 5 
(4) 13
Ans.
(4)
Let,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Given, distance between P1 and P2 is 1/3
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
And distance between P1 and P3 is 2/3
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.56. If the line JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA intersects the plane 2x + 3y - z + 13 = 0 at a point P and the plane 3x + y + 4z = 16 at a point Q, then PQ is equal to:    (2019)
(1) 14
(2) √14
(3) 2√7
(4) 2√14
Ans.
(4)
Let points P (3λ + 2, 2λ - 1, - λ + 1) and Q(3μ + 2, 2μ - 1, - μ + 1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, P (- 1, -3, 2)
Similarly, Q lies on 3x + y + 4z = 16
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, Q is (5, 1, 0)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.57. A plane which bisects the angle between the two given planes 2x - y + 2z -4 = 0 and x + 2y + 2z - 2 = 0, passes through the point:    (2019)
(1) (-1, -4, 1)
(2) (1, 4, -1)
(3) (2, 4, 1)
(4) (2, -4, 1)
Ans.
(4)
The equations of angle bisectors are,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
⇒ x - 3y - 2 = 0
or 3x + y + 4z - 6 = 0
(2, -4, 1) lies on the second plane.

Q.58. The length of the perpendicular drawn from the point (2, 1,4) to the plane containing the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2019)
(1) 3
(2) 1/3
(3) √3
(4) 1/√3
Ans.
(3)
The equation of plane containing two given lines is,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
On expanding, we get x - y - z = 0
Now, the length of perpendicular from (2, 1, 4) to this plane
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA 

Q.59. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAbe a vector coplanar with the vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis equal to:    (2018)
(1) 336
(2) 315
(3) 256
(4) 84
Ans.
(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.60. If L1 is the line of intersect ion of the planes  2x - 2y + 3z = 0, x - y + z = 0 and L2 is the line of intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z = 0 then the distance of the origin fro m the plane, containing the lane L1 and L2 , is:    (2018)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(2)
Vectors along the given lines L1, L2 are
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Putting y = 0 in 1st two equation
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Point on the plane is (–5, 0, 4) and normal vector of required plane is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Hence, equation of plane is - 7x + 7y - 8z - 3 = 0
Perpendicular distance is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.61. The length of the project ion of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = 7 is:    (2018)
(1) 2/√3
(2) 2/3
(3) 1/3
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(4)
D.R’s of AB = (1, 0, 1)
D.R’s of normal to plane = (1, 1, 1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
AB = √2
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Length of projection = JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.62. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAbe a vector such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and the angle between JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAbe 30°. Then JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis equal to    (2017)
(1) 1/8
(2) 25/8
(3) 2
(4) 5
Ans.
 (3)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA 

Q.63. If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line, x/1 = y/4 = z/5 is Q, then PQ is equal to    (2017)
(1) 6√5
(2) 3√5
(3) 2√42
(4) √42
Ans.
(3)
Equation of PQ,
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Let M be (λ+1, 4λ- 2, 5λ+ 3)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
As it lies on 2x + 3y – 4z + 22 = 0
λ = 1
For Q, λ = 2
Distance PQ
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA 

Q.64. The distance of the point (1, 3, –7) from the plane passing through the point (1, –1, –1), having normal perpendicular to both the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is     (2017)
(1) 10/√74
(2) 20/√74
(3) 10/√83
(4) 5/√83
Ans. 
(3)
Let the plane be
a(x - 1)+ b( y + 1) + c(z +1) = 0
It is perpendicular to the given lines
a – 2b + 3c = 0
Solving, a : b : c = 5 : 7 : 3
∴ The plane is 5x + 7y + 3z + 5 = 0
Distance of (1, 3, –7) from this plane = 10/√83 

Q.65. The coodinates of the foot of the perpendicular from the point (1, -2,1) on the plane containing the lines, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2017)
(1) (2, -4, 2)
(2) (1, 1, 1)
(3) (0, 0, 0)
(4) (-1, 2, -`= 1)
Ans.
(3)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.66. The line of intersection of the planes JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2017)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(3)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
13 x = 6
x = 6/13y
y = 5/13
.... is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.67. The area (in sq. units) of the parallelogram whosed diagonals are along the vectors JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2017)
(1) 20
(2) 65
(3) 52
(4) 26
Ans.
(2)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.68. If the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA lies in the plane, 2x - 4y + 3z = 2, then the shortest distance between this line and the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is:    (2017)
(1) 1
(2) 2
(3) 3
(4) 0
Ans.
(4)
pt (3, -2, λ) on pline 2x - 4y +3z - 2 = 0
= 6 + 8 - 3λ - 2 = 0
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ...(2)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
p (1, 0, 0)
gives are ditersech - thortest distance = 0

Q.69. If x = a,  y = b, z = c is a solution of the system of linear equations
x + 8y + 7z = 0
9x + 2y +3z = 0
y + y + z = 0
such that the point (a, b, c) lies on the plane x + 2y + z = 6, then 2a + b + c equals:    (2017)
(1) 2
(2) -1
(3) 1
(4) 0
Ans.
(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.70. If the vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis written as the sum of a vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA parallel to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and a vector JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA perpendicular to JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is equal to:    (2017)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.71. If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of ΔABC is.    (2017)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans.
(1)
Let Centroid be (h, k, l)
∴ x - intp = 3h Y - intp = 3k, 3 - int = 3l
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.72. The distance of the point (1, -5, 9) from the plane x - y + z = 5 measured along the line x = y = z is:    (2016)
(1) 3√10
(2) 10√3
(3) 10/√3
(4) 20/3
Ans.
(2)
Let Q(1, -5, 9)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Line is JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA(say)
Any pt on line we can take P(r + 1, r - 5, r + 9)
So, Pt satisfy Plane
=>(r + 1) - (r - 5) + (r + 9) = 5 r = -10 So, Point P = (-9, -15, -1)
Distance is PQ = JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.73. If the line, JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA lies in the plane, lx + my - z = 9, then l2 + m2 is equal to:    (2016)
(1) 26
(2) 18
(3) 5
(4) 24
Ans. (4) 
Point on line is P = (3, -2, -4)
'P’ lies on lx + my - z = 9
=> 3l - 2m + 4  = 9
3l - 2m = 5 ....(1)
As line lies on plane
=> 2 x l + m x (-1) + 3 x (-1) = 0
2l - m = 3 .....(3) 
Solving l = 1, m = -1
So, l+ m= 2 

Q.74. Let JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAandJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAbe three unit vectors such that JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAIf JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis not parallel to c , then the angle betweenJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAandJEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAis:    (2016)
(1) 3π/4
(2) π/2
(3) 2π/3
(4) 5π/6
Ans.
(4)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Equate,JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA as JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDAunit vectors
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.75. The shortest distance between the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA lies in the interval:    (2016)
(1) (2, 3]
(2) [0, 1)
(3) (3, 4]
(4) [1, 2)
Ans.
(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.76. The distance of the point (1, -2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0, is   (2016)
(1) 1/√2
(2) 2
(3) √2
(4) 2√2
Ans.
(4)
Equation of plane ⊥ to the planes.
x - y + 2z = 3 & 2x - 2y + z + 12 = 0
and passes through (1, 2, 2) is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
3(x - 1) + 3(y - 2) = 0
x + y = 3 ..... (1)
distance of plane x + y - 3 = 0 from (1, - 2, 4) is
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.77. In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA then the point (p, q) lies on a line    (2016)
(1) parallel to y-axis
(2) making an acute angle with the positive direction of x-axis
(3) parallel to x-axis
(4) making an obtuse angle with the position direction of x-axis.
Ans.
(2)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA ⇒ - 8 + 2(9 - 1) - 3 (p + 1) = 0 ⇒ - 3p + 2q - 13 = 0
⇒ (p, q) lies on line
3x - 2y + 13 = 0
slope = 3/2

Q.78. ABC is a triangle in a plane with vertices A(2, 3, 5), B(-1, 3, 2) and C(λ, 5, µ). If the median through A is equally inclined to the coordinate axes, then the value of (λ3 + µ3  + 5) is    (2016)
(1) 676
(2) 1130
(3) 1348
(4) 1077
Ans.
(3)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
direction cosine of AD = JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.79. Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA and JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA respectively, then the position vector of the orthocentre of this triangle, is    (2016)
(1) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(2) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(3) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
(4) JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Ans. 
(4)
Position vector of the centroid of JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
Now we known that centroid divides the line joining orthocentre to circum centre divided by centriod divided by centroid in the ratio in 2 : 1
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

Q.80. The number of distinct real values of λ for which the lines JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA are coplanar is    (2016)
(1) 3
(2) 2
(3) 1
(4) 4
Ans.
(1)
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA
JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA 

The document JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA is a part of the NDA Course Mathematics for NDA.
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237 videos|243 docs|240 tests
237 videos|243 docs|240 tests
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Summary

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JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

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JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

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Important questions

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mock tests for examination

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Previous Year Questions with Solutions

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Free

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Sample Paper

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past year papers

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practice quizzes

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ppt

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Objective type Questions

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JEE Main Previous Year Questions (2016- 2023): Vector Algebra and Three Dimensional Geometry | Mathematics for NDA

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pdf

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