# 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 1=> 1

^{2 }+ 1^{2 }+ 1^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 1=>

=> n

^{2}= 1 – 1=> n

^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 1=> p

^{2 }+ p^{2 }+ p^{2}= 1=> 3p

^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 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,

=> l

^{2 }+ m^{2 }+ n^{2}= 1=>

=>

Hence, the vector is given as,

=>

=>

=>

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