Class 12 Maths Chapter 10 Exercise 10.4

Class 12 Maths Chapter 10 Exercise 10.4 - Vector Algebra NCERT Solutions

Class 12 Maths Chapter 10 Exercise 10.4 covers advanced topics in Vector Algebra. Our NCERT Solutions for Exercise 10.4 offer clear, step-by-step explanations for every problem, helping you build a solid understanding of these concepts. This exercise is an important part of Chapter 10, providing crucial practice for mastering Vector Algebra.

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Q1. Find ,if and .

Solution. We are given that

Q2.Find a unit vector perpendicular to each of the vector and, where and

Solution. We are given vectors and, where

The unit vector that is perpendicular to both vectors = cross product of both vectors/Magnitude of the cross product of both vectors

The unit vector that is perpendicular to both vectors

Q3.If a unit vector makes an angle π/3 with , π/4, with and an acute angle θ with , then find θ and hence ,the components of

Solution: Let the unit vector has the components a₁,a₂ and a₃

Therefore

Since is a unit vector

It is also given that makes an angle π/3 with , π/4, with and an acute angle θ with

We know that

and ,two vectors that makes an angle θ with each other then we have

Similarly, we can get

Now, we have

Squaring both sides

cos²θ + 3/4 =1

cos²θ =1/4

cos θ = 1/2⇒a₃=1/2

Hence components of are 1/2,1/√2 and 1/2

Q4. Show that

Solution. Taking LHS

By the distributive property of the vector product

Q5.Find λ and μ if

Solution. We are given that

Simplifying it

Comparing both sides

6μ – 27λ = 0….(i) ,2μ – 27=0…..(ii), 2λ – 6 = 0…….(iii)

From equation (ii) , (ii)

μ = 27/2, λ = 3 and

Q6.Given that and ,what can you conclude about the vectors and ?

Solution. It is given that

Equation (i) implies that either or cos θ=0(i.e both vectors are perpendicular to each other)

It is also given that

Equation (ii) implies that either or sin θ=0(i.e both vectors are parallel to each other)

From Equation (i) and Equation (ii), it is to be concluded that both vectors can’t be parallel and perpendicular simultaneously.

Therefore

Q7. Let the vectors given as , , , then show that

Solution. The given vectors are , ,

Evaluating LHS

First of all evaluating

Now, simplifying LHS

……..(i)

Simplifying RHS

……(ii)

And

…….(iii)

Adding (ii) and (iii) equations

Hence

Q8. If either or then . Is the converse true?Justify your answer with an example.

Solution. We are given that if either or then .

Let’s prove if then or

Taking two vector and parallel to each other

Let

Hence converse of the given situation is not true.

NCERT solutions for class 12 maths

Chapter 1-Relations and Functions	Chapter 9-Differential Equations
Chapter 2-Inverse Trigonometric Functions	Chapter 10-Vector Algebra
Chapter 3-Matrices	Chapter 11 – Three Dimensional Geometry
Chapter 4-Determinants	Chapter 12-Linear Programming
Chapter 5- Continuity and Differentiability	Chapter 13-Probability
Chapter 6- Application of Derivation	CBSE Class 12- Question paper of maths 2021 with solutions
Chapter 7- Integrals
Chapter 8-Application of Integrals