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Edited By Satyajeet Kumar | Updated on Jan 24, 2022 06:20 PM IST

The RD Sharma class 12 solution of Algebra of Vectors exercise 22.9 is a tough chapter to be solved but not a tricky one. The RD Sharma class 12th exercise 22.9 explains some of the important factors of this chapter and helps you to go further by making previous concepts clear. The Class 12 RD Sharma chapter 22 exercise 22.9 solution consists of a total of 16 level 1 questions to acknowledge the concepts of the chapter.

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Algebra of Vectors Exercise 22.9 Question 1

Given: Can a vector have direction angles

Hint: Verify using

Explanation: Given angles are

Now the cosines of the direction angles are

i.e.

We know if l ,m ,n be the direction cosines of any line then

L.H.S=

=R.H.S

Hence are the direction cosines of the vector having direction angles

So, the given angles can be the direction angles of a vector.

Algebra of Vectors Exercise 22.9 Question 2

Given: Prove that 1, 1, 1 cannot be the direction cosines of a straight line.

Hint: Check by using

Explanation: We know if l, m, n are the direction cosines of any line then

But here

So 1, 1, 1 can’t be the direction cosines for any line.

Algebra of Vectors Exercise 22.9 Question 3

Answer:Given: A vector makes an angle of with each of x-axis and y-axis.

Hint: Using

Explanation: Let be the angle made with x- axis , y- axis and z- axis respectively.

According to given:

l, m, n be the direction cosines ----- (A)

Here

The angle made by the vector with the z-axis is .

Algebra of Vectors Exercise 22.9 Question 4

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

Hint: Find magnitude of

Explanation: Let any vector

For equal angles

Algebra of Vectors Exercise 22.9 Question 5

Given: A vector is inclined at equal acute angles to x-axis at 45

find

Hint: Use

Explanation: Here makes an angle 45

So,

Now we know if l , m , n be the direction cosines then

Therefore

Algebra of Vectors Exercise 22.9 Question 6 (i)

Answer:Given:

Hint: Find

Explanation: Let be the given vector.

Then magnitude of vector is

Let the direction cosines of vector ‘u’ be

We have

We have

We have

The direction cosines of given vector are

Algebra of Vectors Exercise 22.9 Question 6 (ii)

Given:

Hint: Find

Explanation: Let be the given vector.

Then magnitude of vector ‘ ’ is

Let the direction cosines of vector are

We have

We have

We have

The direction cosines of given vector are

Algebra of Vectors Exercise 22.9 Question 6 (iii)

Given:

Hint: Find

Explanation: Let be the given vector.

Then magnitude of vector is

Let the direction cosines of vector are

We have

We have

We have

The direction cosines of given vector are

Algebra of Vectors Exercise 22.9 Question 7 (i)

Given:

Hint: Find

Explanation: Let be the given vector

Then magnitude of vector is

Let the direction angles of vector are

We have

We have

We have

The angles of the given vector are

Algebra of Vectors Exercise 22.9 Question 7 (ii)

Answer:Given:

Hint: Find

Explanation: Let be the given vector

The magnitude of vector is

Let the direction angle of the vector are

We have

We have

We have

The angles of the given vector are

Algebra of Vectors Exercise 22 .9 Question 7 (iii)

Answer:Given:

Hint: Find

Explanation: Let be the given vector

The magnitude of vector is

Let the direction angle of the vector are

We have

We have

We have

The angles of the given vector are

Algebra of Vectors Exercise 22.9 Question 8

Answer: Direction cosines are equalGiven: Show that the vector is equally inclined with the axis OX, OY and OZ.

Hint: Find then direction direction cosines

Explanation: Let

A vector is equally inclined to OX, OY, OZ

i.e. X, Y and Z axis respectively.

If its direction cosines are equal

Direction ratios of are a=1,b=1,c=1

Magnitude of

Direction cosines of

Since the direction cosines are equal

is equally inclined to OX, OY and OZ.

Algebra of Vectors Exercise 22.9 Question 9

Answer: Direction cosines are equally inclined to axis.Given: Show that the direction cosines of a vector equally inclined to the axis OX, OY and OZ are

Hint: Find

Explanation: Let the required vector be

Direction ratios are a, b, c

Since the vector is equally inclined to axis OX, OY and OZ, thus the direction cosines are equal.

Since

The vector is

Magnitude of

Direction cosines are

Hence Proved

Algebra of Vectors Exercise 22.9 Question 10

Given: A unit vector makes an angle withwith and an acute angle with , then find and hence the component of

Hint: Find x, y, z

Explanation: Let us take a unit vector

So magnitude of

Angle with

Angle of with

Also,

Angle of with

Now,

Magnitude of

Squaring on both sides

Since is an acute angle

So,

is in 1st Quadrant

And is positive in 1st Quadrant

So,

And

Hence

The required vector is

So, components of are

Algebra of Vectors Exercise 22.9 Question 11

Answer:Given: Find a vector of magnitude units which makes an angle of angle of and with y and z axis respectively.

Hint: Use given conditions to find

Explanation: let

Squaring on both sides

It makes with y-axis

So,

Also it makes with z-axis

So,

So,

Algebra of Vectors Exercise 22.9 Question 12

Given: A vector is inclined at equal angles to the three axis. If the magnitude of is ,find

Hint: Use

Explanation: Let be the angles inclined to the three axis.

Since is inclined equal angle to axis

So,

RD Sharma class 12th exercise 22.9 is prepared specifically for students who want to gain more knowledge on the core concepts of the subject. To strengthen their basics and score good marks in exams, students can refer to this material. It is prepared by a group of subject experts who have years of experience with CBSE exam papers.

Moreover, RD Sharma class 12 solution chapter 22 exercise 22.9 solutions contain step-by-step answers which the students can refer to if they find any question difficult. As this material is updated to the latest version of the book, students can prepare for their exams and finish their homework.

RD Sharma class 12th exercise 22.9 material is the best alternative for students as it contains questions and answers in one place. Therefore making it easier to revise and prepare. However, as mathematics includes hundreds of questions, it gets difficult for students to remember all concepts.

This material is made specifically to ease the burden on students. They can prepare systematically and quickly come back to any problem which they do not understand. Every answer from this material goes through a series of quality checks to ensure that the best information is available for students.

Additionally, the RD Sharma class 12th exercise 22.9 material is available on Career360's website free of cost. Students can access them through any device with an internet connection. This means that they can use their mobile or laptop to study from their homes.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Are vectors important for boards?

Yes, each and every chapter given in the RD Sharma textbook is definitely important for board exams.

2. Can I skip the algebra of vectors in board exams?

It is always said to consider each and every chapter of RD Sharma to be important for board exams, therefore skipping any chapter will not be a smart move

3. Is RD Sharma solution class 12 helpful for homework?

Yes, it is a benefit for students to use the RD Sharma solutions when solving homework to get help from the solved questions.

4. Are the questions in this exercise of the latest version?

Yes, the solutions that are provided in this exercise are thoroughly revised and updated for better knowledge.

5. Is this material available online?

Yes, it is available on the Careers360 website and also all other RD Sharma solutions can be found on this website.

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