JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - JEE MCQ

# JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - JEE MCQ

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## 46 Questions MCQ Test - JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry

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JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 1

### The scalar  equals :

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 1

if any two vector are equal out of

= 0

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 2

### For  non-zero vectors  holds if and only if

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 2

where θ is angle between

where θ is angle between

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 3

### The volume of the parallelopiped whose sides are given by

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 3

Vol. of parallelopiped =

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 4

The points with position vectors 60i + 3j, 40 i – 8 j, ai – 52 j are collinear if

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 4

Three pts A, B, C are collinear if

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 5

Let   be three non - coplanar vectors and   are vectors defined by the relations   then the value of the expression   is equal to

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 5

Given that   are non coplanar

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 6

Let a, b, c be distinct non-negative numbers. If the vectors  lie in a plane, then c is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 6

a, b, c are distinct non negative numbers and the vectors  are coplanar.

⇒ c2 – ac – ab + ac = 0
⇒ c2 = ab Þ  a, c, b are in G.P.
∴ c is the G.M. of  a and b.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 7

Let  be the position vectors of P and Qr espectively, with respect to O and    The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular then

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 7

We have

[Internal divison]

[external divison]

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 8

Let α, β, γ be distinct real numbers. The points with position vectors

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 8

Let the given position vectors be of point A, B and C respectively, then

⇒ ΔABC is an equilateral Δ.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 9

Let  is a unit vector such that   equals

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 9

where x2 + y2 + z2 = 1 ....(1)

⇒ x – y = 0 ⇒ x = y ...(2)

⇒ x + y + z = 0
⇒ 2x + z = 0 (using (2))
⇒ z = – 2x ...(3) From (1), (2) and (3)

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 10

If  are non coplanar unit vectors such that   then the angle between  is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 10

are non-coplanar]

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 11

Let   be vectors such that   If  and

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 11

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 12

If  are three non coplanar vectors, then  equals

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 12

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 13

Let a = 2i + j – 2k and b = i + j. If c is a vector such that a. c = | c |, | c - a | = 2√2 and the angle between (a × b) and c is 30°, then | (a × b) × c| =

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 13

Substituting values of    in (1), we get

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 14

Let a =2i + j + k, b = i +2j –k and a unit vector c be coplanar. If c is perpendicular to a, then c =

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 14

As c is coplanar with a and b, we take,

c = αa + βb  ...(1)

where α,β are scalars.
As c is perpendicular to a , c.a = 0
∴ From (1) we get, 0 = α a.a + β b.a

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 15

If the vectors  form the sides BC, CA and ABrespectively of a triangle ABC, then

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 15

(by triangle law)

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 16

Let the vectors  be such that   Let P1 and P2 be  planes determined by the pairs of vectors   respectively. Thenthe angle between P1 and P2 is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 16

Given that   are vectors such that

P1 is the plane determined by vectors
∴ Normal vectors  will be given by
Similarly, P2 is the plane determined by vectors
∴ Normal vectors   to P2 will be given by
Substituting the values of

and hence the planes will also be parallel to each other.
Thus angle between the planes = 0

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 17

If  are unit coplanar vectors, then the scalar triple product

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 17

are unit coplanar vectors,   and   a re a lso coplan a r vector s. bein g lin ea r combination of

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 18

and  depends on

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 18

= 1(1+ x - y - x + x2) -1(x2-y) = 1
∴   Depends neither on x nor on y.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 19

If   are unit vectors, then  does NOT exceed

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 19

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 20

are two unit vectors such that  and   are perpendicular to each other then the angle between

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 20

Given that   are two unit vectors

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 21

Let   is a unit vector,, then the maximum value of the scalar triple product

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 21

Given that   and u is a unit vector

which is max. when cos θ =1

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 22

The value of k such that  lies in the plane 2x – 4y + z = 7, is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 22

As the line  lies in th e plan e 2x - 4 y +z= 7, the point (4, 2, k) through which line passes must also lie on the given plane and hence 2 × 4 – 4 × 2 + k = 7  ⇒ k = 7

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 23

The value of ‘a’ so that the volume of parallelopiped formed by  becomes minimum is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 23

Vol. of parallelopiped formed by

= 1(1- 0) - a(0 - a2) +1(0-a) = 1 + a3-a

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 24

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 24

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 25

If the lines

intersect, then the value of k is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 25

⇒ x = 3 + m,y = k +2μ and z = μ
Since above lines intersect
...(1)
3λ - 1 = 2μ + k            ...(2)
μ = 4λ+1                    ..(3)

Solving (1) and (3) and putting the value of  l and m in (2) we get, k = 9/2

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 26

The unit vector which is orthogonal to the vector   an d is coplanar with the vectors   and

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 26

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 27

A variable plane at a distance of the one unit from the origin cuts the coordinates axes at A, B and C. If the centroid D (x, y, z) of triangle ABC satisfies the relation  , then the value k is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 27

Let the eqn of variable plane be   which meets the axes at A (a, 0, 0), B (0, b, 0) and C (0. 0, c).
∴ Centroid of

and it satisfies the relation

...(1)

Also given that the distance of plane  from (0, 0, 0) is 1 unit.

From (1) and (2), we get

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 28

If  are three non-zero, non-coplanar vectors and

then the set of orthogonal vectors is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 28

We observe that

is a set of orthogonal vectors.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 29

A plane which is perpendicular to two planes 2x – 2y + z = 0 and x – y + 2z = 4, passes through (1, –2, 1). The distance of the plane from the point (1, 2, 2) is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 29

The equation of plane through the point (1, –2, 1) and perpendicular to the planes 2 x - 2 y +z=0 and x - y + 2z=4 is given by

It’s distance from the point (1, 2, 2) is

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 30

L et  A vector in the plane of   whose projection on

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 30

A vector in  the plane of   is

Projection of

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 31

The number of distinct real values of λ, for which the vectors  are coplanar, is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 31

We know that three vector are coplanar if their scalar triple product is zero.

∴ Two real solutions.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 32

Let  be unit vectors such that  Which one of the following is correct ?

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 32

Since,   ar e unit vectors ,
therefore   form an equilateral triangle.

Also since  are  non parallel (these form an equilateral Δ).

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 33

The edges of a parallelopiped are of unit length and are parallel to non-coplan ar unit vectors   such that  Then, the volume of the parallelopiped is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 33

We know that th e volume of a par allelopipe with coterminus edges as the vectors   is given by

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 34

Let two non-collinear unit vectors form an acute angle. A point P moves so that at any time t the position vector   (where O is the origin) is given by  When P is farthest from origin O, let M be the length of   and   be the unit vector along

Then,

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 34

Maximum occurs at

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 35

Let P (3, 2, 6) be a point in space and Q be a point on the line

Then the value of m  for which the vector  is parallel to the plane x – 4y + 3z = 1 is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 35

Given that P (3, 2, 6) is a point in space and Q is a point on line

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 36

are unit vectors such that

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 36

are coplanar

So, angle between  should also be π / 3.

Hence are non parallel.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 37

A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 37

The line has +ve and equal direction cosines, these are   or direction ratios are 1, 1, 1. Also thelines passes through P (2, – 1, 2).
∴ Equation  of line is

be a point on this line where it meets the plane 2 x + y + z = 9
Then Q must satisfy the eqn of plane

∴ Q has coordintes (3, 0, 3)

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 38

Let P, Q, R and S be the points on the plane with position vectors  respectively. The quadrilateral PQRS must be a

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 38

⇒ PQRS is a parallelogram but neither a rhombus nor a rectangle.

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 39

Equation of the plane containing the straight line   and perpendicular to the plane containing the
straight lines

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 39

Plane containing two lines  and

Now equation of plane containing the line   and perpendicular to the plane 8x – y – 10 z = 0 is

⇒ –26x +52y –26z = 0 or x –2y+ z = 0

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 40

If the distance of the point P (1, –2, 1) from the plane x + 2y –2z = α, where α > 0, is 5, then the foot of the perpendicular from P to the plane is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 40

As perpendicular distance of x + 2 y –2z = a from the point (1, – 2, 1) is 5

∴ Plane becomes x + 2 y – 2z – 10 = 0

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 41

Two adjacent sides of a parallelogram ABCD are given by

The side AD is rotated by an acute angle a in the plane of the parallelogram so that AD becomes AD¢. If AD¢ makes a right angle with the side AB, then the cosine of the angle a is given by

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 41

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 42

Let be three vector s. A vector the plane of   whose projection on  , is given by

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 42

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 43

The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, –1, 4) with the plane 5x – 4y – z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 43

Equation of st. line join ing Q (2, 3, 5) an d R (1, –1, 4) is

Now let point S on QR be

∵ S is the foot of perpendicular drawn from T (2, 1, 4) to QR,  where dr’s of ST are µ, 4µ – 2, µ –1 and dr’s of QR are  –1, – 4, –1

∴ Distance between P and S

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 44

The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x – y + z = 3 and at a distance  from the point (3, 1 ,–1) is

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 44

The plane passing thr ough the intersection line of given planes is

(x + 2y + 3z – 2) + λ( x -y +z - 3)=0 or (1 +λ)x + (2 -λ)y + (3 +λ)z + (-2 - 3λ) = 0

Its distance from the point (3, 1, –1) is

∴ Required equation of plane is

or 5x – 11y + z = 17

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 45

If   are vectors such that  and   then a possible  value of

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 45

Given that

But neither   is a null vector

JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 46

Let P be the image of the point (3,1,7) with r espect to the plane x – y + z = 3. Then the equation of the plane passing through P and containing the straight line

Detailed Solution for JEE Advanced (Single Correct MCQs): Vector Algebra and Three Dimensional Geometry - Question 46

P, the image of point (3, 1, 7) in the plane x – y + z = 3 is given by

Now equation of plane through (–1, 5, 3) and containing the
line

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