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Edited By Kuldeep Maurya | Updated on Jan 27, 2022 05:07 PM IST

The syllabus for the class 12 students is no less when compared to the other grades. As the challenges get bigger, every student spends on preparing for the exams gets extended. Not everyone who practices reaps the fruit; only the ones who practice in the right way can attain it. Students encounter difficulties in solving sums in the chapters like the Scalar and Dot Product. RD Sharma Solutions And as a solution for all their doubts, the RD Sharma Class 12th Exercise 23.1 plays a major role.

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Scaler or Dot Products Exercise 23.1 Question 1(i)

Scaler or Dot Products Exercise 23.1 Question 1(ii)

Scaler or Dot Products Exercise 23.1 Question 1(iii)

Answer: 5Hints: you must know the property of vectors.

Given: and

Solution:

Scaler or Dot Products Exercise 23.1 Question 2(i)

and

If * them both are perpendicular to each other *

Scaler or Dot Products Exercise 23.1 Question 2(ii)

If them both and are to each other

Scaler or Dot Products Exercise 23.1 Question 2(iii)

If them both and are perpendicular to each other

Scaler or Dot Products Exercise 23.1 Question 2(iv)

If them both and are perpendicular to each other

Scaler or Dot Products Exercise 23.1 Question 3

Answer:Hints: you must know properly of finding angle from vectors.

Given: and

Solution:

Now, we know

Scaler or Dot Products Exercise 23.1 Question 4

Answer: - 9Hint: you must know the rules of solving vectors.

Given: and

Solution: and

Find

Scaler or Dot Products Exercise 23.1 Question 5(i)

Now we know * *

* *

* *

Scalar or Dot Products Exercise 23.1 Question 5(ii)

We know

Now,

Magnitude of

Magnitude of

Put values in 1

Scalar or Dot Products Exercise 23.1 Question 5(iii)

We know

Now,

Magnitude of

Magnitude of

=

Put in 1

Scalar or Dot Products Exercise 23.1 Question 5(iv)

we Know ....(i)

Now,

Magnitude of

=

Magnitude of

=

Put in (1)

Scalar or Dot Products Exercise 23.1 Question 5(v)

=

Put in 1

Scalar or Dot Products Exercise 23.1 Question 6

(Because it is a unit vector along x-axis)

Now, Let * be the angle between ** and y-axis.*

(become it is a unit vector along* **y**-**axis**)*

Now, Let * be the angle between **and y-axis.*

Scalar or Dot Products Exercise 23.1 Question 7(i)

Give that,

From 1 and 2, subtract

From 2 and 3, subtract

Where

Put value of (a) and (c) in 1

Now put value of b in (c) and (a)

and

So,

Scalar or Dot Products Exercise 23.1 Question 7(ii)

Let be the required vector.

Given that

From 1 and 2 subtract

From 1 and 2 subtract

From 1 and 2 subtract

Put b = -1

Now from 3

Put value of b and c in 1

Scalar or Dot Products Exercise 23.1 Question 8(i)

Answer: provedHint: you must know the rules of solving vectors.

Given: and are unit vector in dined at angle then prove

Solution: given that and are unit vectors

We have,

Now,

Scalar or Dot Products Exercise 23.1 Question 8(ii)

So

We have,

And

And similarly

Scalar or Dot Products Exercise 23.1 Question 9

Now

Scalar or Dot Products Exercise 23.1 Question 10

Answer: provedHint: you must know the rules of proving vector

Given: if are these mutually perpendicular unit vectors, prove that

Solution: given that and are unit vectors

So,

Since they are mutually perpendicular

Now,

=Proved

Scalar or Dot Products Exercise 23.1 Question 12

1

we Know

Again , Let * be the angle between ** and z-axis*

* *

*We Know *

* *

* *

* *

From 1, 2, 3 the given vector is equally inclined to the co-ordinate ones.

Scalar or Dot Products Exercise 23.1 Question 13

And

So,

And

Therefore, the given vectors are mutually perpendicular unit vectors.

Scalar or Dot Products Exercise 23.1 Question 14

Solution: we have

Proved

Scalar or Dot Products Exercise 23.1 Question 15

Solution: The given vectors are

and

Now,

It is given that

Scalar or Dot Products Exercise 23.1 Question 16

Then find the value of , so and are perpendicular vectors

Given that is orthogonal to* *

Scalar or Dot Products Exercise 23.1 Question 17

Then express in the form of where is parallel to and is perpendicular to

Also,

Since *parallel to *

Substituting the value of * and *

* *

Since * is to ** *

we get

Scalar or Dot Products Exercise 23.1 Question 18

But the inverse need not be true, justify with example

or

Then,

Now let us assume that

But here we cannot say that either or

For example, Let,

Scalar or Dot Products Exercise 23.1 Question 19

form a right angle

Scalar or Dot Products Exercise 23.1 Question 20

Solution: We have,

Scalar or Dot Products Exercise 23.1 Question 21

Scalar or Dot Products Exercise 23.1 Question 22

Also,

we Know

Scalar or Dot Products Exercise 23.1 Question 23

from a right triangle.

So, is perpendicular to

So, is a right angle triangle.

Scalar or Dot Products Exercise 23.1 Question 24

Scalar or Dot Products Exercise 23 .1 Question 25

Now,

So, * is perpendicular to ** *

So, *is right angled at C. *

Scalar or Dot Products Exercise 23.1 Question 26

Answer: 2Hint: You must know the rules of solving vectors.

Given: Find the projection of on , where ,

Solution: Given that,

and

Projection of* * on* * * is*

* *

Scalar or Dot Products Exercise 23.1 Question 27

And

Now,

So,is orthogonal of

Scalar or Dot Products Exercise 23.1 Question 28

Now,

Again,

Now from (1)

NOw,

And,

Scalar or Dot Products Exercise 23.1 Question 29

Answer:Hint: You must know the rules of solving vectors.

Given: If two vectors and are such that and then find the value of

Solution: Given that,

and

Now,

Scaler or Dot Products Exercise 23.1 Question 30(i)

Scaler or Dot Products Exercise 23.1 Question 30(ii)

Scaler or Dot Products Exercise 23.1 Question 31(i)

Scaler or Dot Products Exercise 23.1 Question 31(ii)

Scaler or Dot Products Exercise 23.1 Question 31(iii)

Scalar or Dot Products Exercise 23.1 Question 32(i)

Answer:Hint: You must know the rules of solving vectors.

Given: Find if , , &

Solution: , &

We know that,

Scaler or Dot Products Exercise 23.1 Question 32(ii)

Scalar or Dot Products Exercise 23.1 Question 32(iii)

, &

We know,

Scalar or Dot Products Exercise 23.1 Question 33(i)

, &

We know that,

Scalar or Dot Products Exercise 23.1 Question 34

and

Let and be such that

Since, is parallel to

Substituting the values, and

Since is perpendicular to

Scalar or Dot Products Exercise 23.1 Question 35

Now,

Scalar or Dot Products Exercise 23.1 Question 36

Let,

And and be such that,

Since is parallel to

Substituting the values of *and *

* *

Since y is perpendicular to

* *

* *

So,

* *

* *

Scalar or Dot Products Exercise 23.1 Question 37

Answer:Hint: You must know the rules of solving vectors.

Given: Decompose the vector into vector which are parallel and perpendicular to the vector

Solution:

Let

And and be such that,

Since

Substituting the values of

Since y is perpendicular to

Scalar or Dot Products Exercise 23.1 Question 38

Given that,

And

Give that borthogonal

Scalar or Dot Products Exercise 23.1 Question 39

Given that

Also given that,

So, it means that for any vector the given equation is satisfied

Scalar or Dot Products Exercise 23.1 Question 40

and

Now,

So, is perpendicular to

Again,

So, is perpendicular to

Scalar or Dot Products Exercise 23.1 Question 41

**Hint**: You must know the rules of solving vectors.**Given**: If

Prove**Solution**

Scalar or Dot Products Exercise 23.1 Question 42

Given that

So, either or

Similarly,

So, or

Also,

So, or

But cannot be perpendicular to as are non-coplanar

So, , is null vector.

Scalar or Dot Products Exercise 23.1 Question 43

and

Now, let be any vector in plane of and

Then, is the linear combination of and

Scalar or Dot Products Exercise 23.1 Question 44

Scalar or Dot Products Exercise 23.1 Question 45

Scalar or Dot Products Exercise 23.1 Question 46

Let be the angle between and

be the angle between and

Given that is acute and is obtuse

Scalar or Dot Products Exercise 23.1 Question 48

Answer:Hint: You must know the rules of solving vectors.

Given: If and are two non-collinear unit vectors such that .Find

Solution: We have,

Squaring both sides

Now,

Scalar or Dot Products Exercise 23.1 Question 49

Squaring both sides,

Now,

So, is perpendicular to .

Scalar or Dot Products Exercise 23.1 Question 50

c and d are perpendicular to each other,

Angle between a and b

So, angle between a and b is

Chapter 23 of class 12 mathematics consists of only two exercises, ex 23.1 and ex 23.2. The first exercise, ex 23.1, has around 67 questions, including its subparts in the textbook. This exercise revolves around the projection of vectors, unit and position vector, the angle between two vectors, dot product, etc. The best reference book to find the answers to all these questions is the RD Sharma Class 12 Chapter 23 Exercise 23.1. Students and the teachers can use this book to clarify their doubts and get to know the various methods in which each of these sums can be solved.

Regular practice is also a major requirement when it comes to developing knowledge in a particular concept. And it is a must to have clarity in mathematics. Even though this single exercise consists of 67 questions, additional sums are required for the students to understand each concept and method in-depth. The RD Sharma Class 12th Exercise 23.1 contains a lot of additional questions for practice. These extra sets of solved sums act as a catalyst in making the students understand the concepts easily.

The Class 12 RD Sharma Chapter 23 Exercise 23.1 Solution reference book is nothing but a collection of accurate solutions by mathematical experts. Not any random person has given in the solutions. You can even check out the reviews about these books given by the previous batch of students. Everyone has benefited by scoring high marks in the public exams using the RD Sharma Class 12 Scalar and Dot Product Solutions Ex 23.1 book.

This book is nothing but a boon for the students who are unable to afford a good set of solution books to prepare for their exams. As the Career 360 website provides all the RD Sharma books and the RD Sharma Class 12th Exercise 23.1 reference book for free of cost, the students prefer no other reference books.

The advantages that a student acquires by using the RD Sharma Class 12 Solutions Chapter 23 Ex 23.1 are ineffable. This will benefit them in every possible way to make them score hight than their benchmark. Download the RD Sharma book now to begin your practice in the right way.

- 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. Which is the best solution book to learn the basics of the Scalar and Dot product concept?

The RD Sharma Class 12th Exercise 23.1 is the best mathematics solution material for the students to understand the basic and in-depth concepts in the Scalar and Dot Products chapter.

2. Where is the complete set of RD Sharma solution books available?

The complete authorized set of RD Sharma books are available on the Career 360 website.

3. How many questions are asked in the first exercise of Chapter 23 in class 12 mathematics?

There are 67 questions asked in Exercise 23.1 in the class 12 mathematics book. The right solutions for these questions can be found easily using the RD Sharma Class 12th Exercise 23.1 solution material.

4. How much should I pay to own the RD Sharma books from the Career 360 website?

All the RD Sharma books available at this site can be accessed for free. Therefore, you need not pay even a single rupee to use these books.

5. Does the Career 360 website provide the feature to download the RD Sharma books?

The Download option can be enabled to save any RD Sharma books from the Career 360 website to your device.

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