# Class 12 Maths NCERT Solutions for Chapter 10 Vector Algebra Exercise 10.2

### Vector Algebra Exercise 10.2 Solutions

**1. Compute the magnitude of the following vectors : **

**Solution**

The given vectors are :

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

**Solution**

Hence, a⃗ and b⃗ are two different vectors having the same magnitude. The vectors are different because they have different directions.

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

**Solution**

The direction cosines of p⃗ and q⃗ are the same. Hence, the two vectors have the same direction.

**4. Find the values of x and y so that the vectors 2i ^ + 3j ^ and xi ^ + yj ^ are equal.**

**Solution**

The two vectors 2i ^ + 3j ^ and xi ^ + yj ^ will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

**5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).**

**Solution**

The vector with the initial point P(2, 1) and terminal point Q(-5, 7) can be given by,

Hence, the required scalar components are -7 and 6 while the vector components are -7i ^ and 6j ^.

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

**Solution**

The given vectors are

**7. Find the unit vector in the direction of the vector a⃗ = i ^ + j ^ + 2k^ .**

**Solution**

The unit vector a ^ in the direction of vector a⃗ = i ^ + j ^ + 2k^ is given by a ^ = a⃗ /|a| .

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

**Solution**

The given points are P(1, 2, 3) and Q(4, 5, 6).

**Solution**

**10. Find a vector in the direction of vector = 5i ^ - j ^ + 2k^ which has a magnitude of 8 units .**

**Solution**

**11. Show that the vectors**

**2i ^ - 3j ^ + 4k^**

**are collinear.**

**Solution**

**12. Find the direction cosines of the vector i ^ + 2j ^ + 3k^ .**

**Solution**

**13. Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.**

**Solution**

The given points are A(1, 2, -3) and B(-1, -2, 1).

**14. Show that the vector i ^ + j ^ + k^ .is equally inclined to the axes OX, OY, and OZ.**

**Solution**

Therefore, the direction cosines of

Now, let Î±, Î² and Î³ be the angles formed by a⃗ with the positive directions of x, y, and z axes.

Now, let Î±, Î² and Î³ be the angles formed by a⃗ with the positive directions of x, y, and z axes.

Then, we have cos Î± = 1/√3 , cosÎ² = 1/√3, cos Î³ = 1/√3 .

Hence, the given vector is equally inclined to axes 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 i ^+ 2j ^ - k^ and -i ^+ j ^ + k^ respectively, in the ratio 2:1.**

(i) Internally

(ii) Externally

(i) Internally

(ii) Externally

**Solution**

The position vector of point R dividing the line segment joining two points P and Q in the ratio m : n is given by :

(i) Internally :

(ii) Externally :

Position vectors of P and Q are given as :

**16. Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, –2).**

**Solution**

The position vector of mid - point R of the vector joining points P(2, 3, 4) and Q(4, 1, -2) is given by,

**17. Show that the points A , B and C with position vectors**

**respectively form the vertices of a right angled triangle.**

**Solution**

Position vectors of points A, B, and C are respectively given as :

Hence, ABC is a right - angled triangle.

Hence, ABC is a right - angled triangle.

**18. In triangle ABC which of the following is not true:**

**Solution**

On applying the triangle law of addition in the given triangle, we have :

From equations (1) and (3), we have :

Hence, the equation given in alternative C is incorrect.

The correct answer is C.

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

(A) b⃗ = Î»a⃗ for some scalar Î» .

(B) a⃗ = ± b⃗

(C) the respective components of a⃗ and b⃗ are not proportional.

(A) b⃗ = Î»a⃗ for some scalar Î» .

(B) a⃗ = ± b⃗

(C) the respective components of a⃗ and b⃗ are not proportional.

**(D) both the vectors a⃗ and b⃗ have the same direction, but different magnitudes.**

**Solution**

If a⃗ and b⃗ are two collinear vectors, then they are parallel.

Therefore, we have :

b⃗ = Î»a⃗ (for some scalar Î»)

Thus, the respective components of a⃗ and b⃗ are proportional.

Thus, the respective components of a⃗ and b⃗ are proportional.

However, vectors a⃗ and b⃗ can have different directions.

Hence, the statement given in D is incorrect.

The correct answer is D.