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

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). 

9. For given vectors,  find the unit vector in the direction of the vector a⃗ + b⃗ .

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

Hence, the given vectors are collinear 

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 α, β and γ be the angles formed by a⃗ with the positive directions of x, y, and z axes. 

Then, we have cos α = 1/√3 , cosβ = 1/√3, cos γ = 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 

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 : 

(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2: 1 is given by, 

(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2 : 1 is given by, 

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. 

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 = λa⃗  for some scalar λ . 
(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  = λa⃗  (for some scalar λ) 

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.