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Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA PDF Download

Q. 1. From a point O inside a triangle ABC, perpendiculars OD, OE, OF are drawn to the sides BC, CA, AB respectively. Prove that the perpendiculars from A, B, C to the sides EF, FD, DE are concurrent.            (1978)

Solution. Let with respect to O, position vectors of points A, B, C, D, E, F be  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Let perpendiculars from A to EF  and from B to DF meet each other at H. Let position vector of H be  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA we join CH.
In order to prove the statement given in question, it is sufficient to prove that CH is perpendicular to DE.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Adding (4) and (5), we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

(using (1), (2) and (3))

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Q. 2. A1, A2,...................... An are the vertices of a regular plane polygon with n sides and O is its centre. Show that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA         (1982 - 2 Marks)

Solution.  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA all vectors are of same magnitude, say ‘a’  and angle between any two consecutive vector  is same  that is Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA be the unit vectors ⊥ to the plane of the polygon.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 3. Find all values of λ such that x, y,z, ≠ (0, 0, 0) and Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDASubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA  are unit vectors along the coordinate axes.             (1982 - 3 Marks)

Ans. λ = 0, - 1

Solution.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
All the above three equations are satisfied for x, y, z not all zero if

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 4. A vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA has components A1, A2, A3 in a right -handed rectangular Cartesian coordinate system oxyz. The coordinate system is rotated about the x-axis through an angle Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA Find the components of A in the new coordinate system, in terms of A1, A2, A3.

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Solution. Since vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA has components A1 , A2 ,A3, in the coordinate system OXYZ,

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

When given system is rotated through Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA  the new x-axis is along old y-axis and new y-axis is along the old negative x-axis z remains same as before.
Hence the components of A in the new system are

A2 , - A1,A3 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 5. The position vectors of the points A, B, C an d D are Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDArespectively. If the points A, B, C and D lie on a plane, find the value of λ .

Ans. 146/17

Solution.Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

We know that A, B, C, D lie in a plane if  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA are coplanar i.e.  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

 
Q. 6. If A, B, C , D are any four points in space, prove that –

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA(area of triangle ABC)

Solution. Let the position vectors of points A, B, C, D be a, b, c, and d respectively with respect to some origin O.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Then,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDASubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA             …(1)
Also Area of ΔABC is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA       … (2)

From (1) and (2), we ge

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Q. 7. Let OA CB be a parallelogram with O at the origin and OC a diagonal. Let D be the midpoint of OA. Using vector methods prove that BD and CO intersect in the same ratio. Determine this ratio.

Solution. OACB is a parallelogram with O as  origin. Let with respect to O position vectors of A and B be  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA respectively..
Then p.v. of C is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Also D is mid pt. of OA, therefore position vector of D is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

CO and BD intersect each other at P.

Let P divides CO in the ratio λ : 1 and BD in the ratio μ : 1 Then by section theorem, position vector of pt. P dividing CO in ratio

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA             …(1)
And position vector of pt. P dividing BD in the ratio μ : 1 is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA              …(2)

As (1) and (2) represent the position vector of same point, we should have

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Equating the coefficients of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA  we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA           … (i)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA               …(ii)

From (ii) we get λ = μ ⇒ P divides CO and BD in the same ratio.

Putting λ = μ in eq. (i) we get μ = 2

Thus required ratio is 2 : 1.


Q. 8. If vectors Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA are coplanar, show that

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Solution. Given that Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA are  three coplanar vectors.

∴ There exists scalars x, y, z, not all zero, such that

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA               … (1)

Taking dot product of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA we get 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA             … (2)

Again taking dot product of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA            … (3)

Now equations (1), (2), (3) form a homogeneous system of equations, where x, y, z are not all zero.

∴ system must have non trivial solution and for this, determinant of coefficient matrix should be zero

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                Hence Proved.

Q. 9. In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, determine the ratio OP : PD using vector methods.

Solution. With O as origin let  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA be the position vectors of A and B respectively.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Then the position vector of E, the mid point of OB is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Again since AD : DB = 2 : 1, the  position vector of D is 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

∴ Equation of OD is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                …(1)

and  Equation of AE is 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA            …(2)

If OD and AE intersect at P, then we will have identical values of Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA Hence comparing the coefficients of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA we get 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Putting value of t in eq. (1) we get position vector of point of intersection P as

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                    … (3)

Now if P divides OD in the ratio λ : 1, then p.v. of P is

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                … (4)

From (3) and (4) we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 10. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDASubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA Determine a vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA Satisfying  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Solution. We are given that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA and  to determine a vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA such that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA and  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

⇒ y -z =-10 … (1)
z -x =-11 … (2)
x -y= 7 … (3)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

⇒ 2x +z=0 … (4)

Substituting y =x- 7 and z = -2x from (3) and (4) respectively in eq. (1) we get

x - 7 + 2x = -10 ⇒ 3x =-3

⇒ x =-1 , y =-8 and z = 2

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 11. Determine the value of ‘c’ so that for all real x, the vector Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA  make an obtuse angle with each other.

Ans. Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Solution. We have,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Now we know that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

As angle between  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA is obtuse, therefore 

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 12. In a triangle ABC, D an d E are points on BC and AC respectively, such that BD = 2 DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PE using vector methods.

Ans. 8 : 3

Solution. Let  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA be the position vectors of pt A, B and C respectively with respect to some origin.

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

ATQ, D divides BC in the ratio 2 : 1 and E divides AC in the ratio 3 : 1.

∴ position vector of D is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA and position vector of E is  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Let pt. of intersection P of AD and BE divides BE in the ratio k : 1 and AD in the ratio m : 1, then position vectors of P in these two cases are
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Equating the position vectors of P in two cases we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA             … (1)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                   … (2)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Dividing (3) by (2) we get

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA   the req. ratio is 8 : 3.


Q. 13. If the vectors  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA are not coplanar, then prove that the vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDASubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA parallel to  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Solution. Given that  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA are  not coplanar Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Consider,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDASubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Here,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                 …(2)

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA                …(3)

[NOTE :  Here we have tried  to write the given expression in such a way that we can get terms involving Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA other terms similar which can get cancelled.]

Adding (1), (2) and (3), we get given vector  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

⇒ given vector = some constant multiple of  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

⇒ given vector is parallel to  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 14. The position vectors of the vertices A, B an d C of a tetrahedron ABCD are Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA respectively. The altitude from vertex D to the opposite face ABC meets the median line through A of the triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA find the position vector of the point E for all its possible positions.

Ans. (-1, 3, 3) or (3, - 1, - 1)

Solution. We are given AD = 4

Volume of tetrahedron  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

We have to find the P.V. of point E.  Let it divides median AF in the ratio λ : 1

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA            …(3)
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA


Q. 15. If A, B and C are vectors such that | B | = | C |. Prove that [(A + B) × (A + C)] × (B × C) (B + C)  = 0 .

Solution. We have,  Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDASubjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA
[∵ (a x b) x c = (a.c)b- (b.c)a]

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

[∵ [A B C] = 0 if any two of A, B, C are equal.]

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Thus, LHS of the given expression

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA

Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA          [∵| B | = C|]

The document Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 | JEE Advanced | Mathematics for NDA is a part of the NDA Course Mathematics for NDA.
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FAQs on Subjective Type Questions: Vector Algebra and Three Dimensional Geometry - 1 - JEE Advanced - Mathematics for NDA

1. What is a vector in mathematics?
Ans. A vector is a mathematical object that has both magnitude and direction. It is represented by an arrow with a specified length and direction. In three-dimensional space, a vector is typically represented as an ordered triple of numbers (x, y, z), where x, y, and z are the components of the vector in the x, y, and z directions, respectively.
2. How are vectors added and subtracted?
Ans. Vectors can be added and subtracted using the parallelogram law of vector addition. To add two vectors, we place them head to tail and draw a parallelogram using the two vectors as adjacent sides. The diagonal of the parallelogram represents the sum of the two vectors. To subtract two vectors, we add the negative of the vector being subtracted.
3. What is the dot product of two vectors?
Ans. The dot product of two vectors is a scalar quantity that measures the degree of similarity or orthogonality between the two vectors. It is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. Mathematically, the dot product of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is given by A · B = a1b1 + a2b2 + a3b3.
4. What is the cross product of two vectors?
Ans. The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It is defined as the product of the magnitudes of the two vectors and the sine of the angle between them, multiplied by a unit vector perpendicular to the plane containing the two vectors. Mathematically, the cross product of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is given by A × B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
5. How can vectors be used in three-dimensional geometry?
Ans. Vectors play a crucial role in three-dimensional geometry. They can be used to represent points, lines, and planes in space. By using vector equations, we can determine the position, direction, and distance between objects in three-dimensional space. Vectors also help in solving problems related to distance, displacement, velocity, acceleration, and forces in three-dimensional systems.
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