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.

Textbook | NCERT |

Class | Class 12th |

Subject | Maths |

Chapter | 23.9 |

Exercise | 23.9 |

Category | RD Sharma Solutions |

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

**Question 1: Ten cards numbered 1 through 10 are placed in a box, mixed **

### 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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