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**Download NCERT solutions for Vector Algebra as PDF.**

## 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

**6. Find the sum of the vectors: **** **** and **** **

**Ans. **Given: and

Adding,

=

**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

=

=

#### NCERT Solutions class 12 Maths Exercise 10.2

**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 =

=

#### NCERT Solutions class 12 Maths Exercise 10.2

**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)

Adding eq. (i) and (ii),

+ = = = [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} = AB^{2} + AC^{2}

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

#### NCERT Solutions class 12 Maths Exercise 10.2

**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),

#### NCERT Solutions class 12 Maths Exercise 10.2

**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

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