Class 12 RD Sharma Solutions – Chapter 23 Algebra of Vectors – Exercise 23.9

Question 1: Can a vector have direction angles 45°, 60°, and 120°.

Solution:

We know that if l, m and n are the direction cosines and ,  and  are the direction angles then,

=> 

=> 

=> 

Also,

=> l² + m² + n² = 1

=> 

=> 

=> As LHS = RHS, the vector can have these direction angles.

Question 2: Prove that 1,1 and 1 can not be the direction cosines of a straight line.

Solution:

Given that, l=1, m=1 and n=1.

We know that,

=> l² + m² + n² = 1

=> 1² + 1² + 1² = 1

=> 3 ≠ 1

Thus, 1, 1 and 1 can never be the direction cosines of a straight line.

=> Hence proved.

Question 3: A vector makes an angle of  with each of x-axis and y-axis. Find the angle made by it with the z-axis.

Solution:

We know that if l, m and n are the direction cosines and ,  and  are the direction angles then,

=> 

=> 

Let  be the angle we have to calculate.

We know that,

=> l² + m² + n² = 1

=> 

=> n² = 1 – 1

=> n² = 0

=> 

=> 

=> 

=>

Question 4: A vector is inclined at equal acute angles to x-axis, y-axis and z-axis. If  = 6 units, find .

Solution:

Given that 

=> 

=> l = m = n = p (say)

We know that,

=> l² + m² + n² = 1

=> p² + p² + p² = 1

=> 3p² = 1

=> 

The vector  can be described as,

=> 

=> 

=>

Question 5: A vector  is inclined to the x-axis at 45° and y-axis at 60°. If  units, find .

Solution:

Given that  and 

We know that,

=> l² + m² + n² = 1

=> 

=> 

=> 

=> 

=> 

=> 

The vector  can be described as,

=> 

=> 

=>

Question 6: Find the direction cosines of the following vectors:

(i): 

Solution:

The direction ratios are given as 2, 2 and -1.

Direction cosines are given as,

=> 

=> 

=>

(ii): 

Solution:

The direction ratios are given as 6, -2 and -3.

Direction cosines are given as,

=> 

=> 

=>

(iii): 

Solution:

The direction ratios are given as 3, 0 and -4.

Direction cosines are given as,

=> 

=> 

=>

Question 7: Find the angles at which the following vectors are inclined to each of the coordinates axes.

(i): 

Solution:

The given direction ratios are: 1,-1,1.

Thus,

=> 

=> 

=> 

=> 

=>

(ii): 

Solution:

The given direction ratios are: 0,1,-1.

Thus,

=> 

=> 

=> 

=> 

=> 

=>

(iii): 

Solution:

The given direction ratios are: 4, 8, 1.

Thus,

=> 

=> 

=> 

=> 

=>

Question 8: Show that the vector  is equally inclined with the axes OX, OY and OZ.

Solution:

Let 

Thus, 

=> 

Thus the direction cosines are: ,  and 

=> 

Thus,

=>

=> Thus, the vector is equally inclined with the 3 axes.

Question 9: Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ,, .

Solution:

Let the vector be equally inclined at an angle of .

Then the direction cosines of the vector l, m, n are: ,  and 

We know that,

=> l² + m² + n² = 1

=> 

=> 

=> 

=> Thus the direction cosines are: , , .

Question 10: If a unit vector  makes an angle  with ,  with  and an acute angle  with, then find \theta and hence the components of .

Solution:

The unit vector be,

=> 

=> 

Given that  is unit vector,

=> 

=> 

=>

=>  

=> 

=> 

=> 

=> 

=> 

=> 

Question 11: Find a vector of magnitude  units which makes an angle of  and  with y and z axes respectively.

Solution:

Let l, m, n be the direction cosines of the vector .

We know that,

=> l² + m² + n² = 1

=> 

=> 

=> 

=> 

Thus vector is,

=> 

=> 

=>

Question 12: A vector  is inclined at equal angles to the 3 axes. If the magnitude of is , find .

Solution:

Let l, m, n be the direction cosines of the vector .

Given that the vector is inclined at equal angles to the 3 axes.

=> 

We know that,

=> l² + m² + n² = 1

=> 

=>

Hence, the vector is given as,

=> 

=> 

=>