# NCERT Solutions class 12 Maths Exercise 10.2

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## NCERT Solutions class 12 Maths Vector Algebra

1. Compute the magnitude of the following vectors:

Ans. Given:

And

Also

#### 2. Write two different vectors having same magnitude.

Ans. Let  and

Clearly,  [ Coefficients of  and  are same in vectors  and  coefficients of  in  and  are unequal as  ]

But

And

#### 3. Write two different vectors having same direction.

Ans. Let  and  =

where  = 2 > 0

Vectors  and  have the same direction.

But

#### 4. Find the values of  and  so that the vectors  and  are equal.

Ans. Given:

Comparing coefficients of  and  on both sides, we have,

and

#### 5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point

Ans. Let  be the vector with initial point A (2, 1) and terminal point B

Position vector of point A is (2, 1) =  and position vector of point B is  =

= Position vector of point B – Position vector of point A

=

=

=

Scalar components of the vectors  are coefficients of  and  in  i.e.,  and 6 and vector components of the vector  are  and

Ans. Given:  and

=

#### 7. Find the unit vector in the direction of the vector

Ans. We know that a unit vector in the direction of the vector  is

8. Find the unit vector in the direction of the vector  where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.

Ans.  Given: Points P (1, 2, 3) and Q (4, 5, 6)

Position vector of point P =  =  and position vector of Q =  = , where O is the origin.

= Position vector of Q – Position vector of P =

=    =

Therefore, the unit vector in the direction of vector  =

=

=

#### 9. For given vectors  and  find the unit vector in the direction of

Ans. Given: Vectors  and

=

#### 10. Find the vector in the direction of vector  which has magnitude 8 units.

Ans. Let

A vector in the direction of vector  which has magnitude 8 units =

=  =

=

#### 11. Show that the vectors  and  are collinear.

Ans. Let  and  =  =

=  where

#### 12. Find the direction cosines of the vector

Ans. The given vector is

=

We know that the direction cosines of a vector  are coefficients of  in  i.e.,

#### 13. Find the direction cosines of the vector joining the points A and B directed from A to B.

Ans. Given: Points A and B

Position vector of point A =  =

And Position vector of point B =  =

Vector  =    =    =

Now  =

A unit vector along

=

Therefore, the direction cosines of vector  =

#### 14. Show that the vector  is equally inclined to the axes OX, OY and OZ.

Ans. Let

Let us find angle  (say) between vector  and OX

=  =

Similarly angle  (say) between vector  and OY  is

And angle  (say) between vector  and OZ  is

=  =

Vector  is equally inclined to OX, OY and OZ.

#### 15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  and  respectively, in the ratio 2 : 1 (i) internally (ii) externally.

Ans. Position vector of P is  and Position vector of Q is

(i) Position vector of point R dividing PQ internally (i.e., R lies within the segment PQ) in the ratio 2 : 1 =  = PR : QR is

=

=

(ii) Position vector of point R dividing PQ externally (i.e., R lies outside the segment PQ and to the right of point Q because ratio 2 : 1 > 1 i.e., PR : QR = 2 : 1) is

=

=

#### 16. Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q

Ans. Given: Point P (2, 3, 4) and Q

Position vector of point P is

And Position vector of point Q is

And Position vector of mid-point R of PQ is  =

=

#### 17. Show that the points A, B and C with position vectors  and  respectively form the vertices of a right angled triangle.

Ans. Given: Position vector of point A is

Position vector of point B is  and

Position vector of point C is  where O is the origin.

Now  = Position vector of point B – Position vector of point A

=  =

=  ……….(i)

= Position vector of point C – Position vector of point B

=  =

=  ……….(ii)

= Position vector of point C – Position vector of point A

=  =

=  ……….(iii)

+  =   =  = [By eq. (iii)]

Now From eq. (i),AB =

From eq. (ii),BC =

From eq. (iii),AC =

Here, we can observe that (Longest side BC)2 =  AB2 + AC2

Therefore, A B, C are the vertices of a right angled triangle.

#### 18. In triangle ABC (Fig. below), which of the following is not true:

(A)

(B)

(C)

Ans. We know by Triangle law of Addition of vectors that

Therefore option (C) is not true because in option (C),

#### 19. If  and  are two collinear vectors, then which of the following are incorrect:

(A)  =  for some scalar

(B)

(C) The respective components of  and  are proportional.

(D) Both the vectors  and  have same direction, but different magnitudes.

Ans. Option (D) is not true because two collinear vectors can have different directions and also different magnitudes.

The option (A) and option (C) are true by definition of collinear vectors.

Option (B) is a particular case of option (A) taking

## NCERT Solutions class 12 Maths Exercise 10.2

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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