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

Here we provide RD Sharma Class 12 Ex 23.9 Solutions Chapter 23 Algebra of Vectors for English medium students, Which will very helpful for every student in their exams. Students can download the latest RD Sharma Class 12 Ex 23.9 Solutions Chapter 23 Algebra of Vectors book pdf download. Now you will get step-by-step solutions to each question.

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

### 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+ m+ 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,

=> l+ m+ n2 = 1

=> 1+ 1+ 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,

=> l+ m+ 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,

=> l+ m+ n2 = 1

=> p+ p+ 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,

=> l+ m+ 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,

=> l+ m+ 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,

=> l+ m+ 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,

=> l+ m+ n2 = 1

=>

=>

Hence, the vector is given as,

=>

=>

=>

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