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

• Last Updated : 28 Mar, 2021

### 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,

=>

=>

=>

### (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,

=>

=>

=>

### (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,

=>

=>

=>

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