Contents

These Solutions are part of NCERT Solutions for Class 12 Maths . Here we have given NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 12 |

Subject |
Maths |

Chapter |
Chapter 10 |

Chapter Name |
Vector Algebra |

Exercise |
Ex 10.1, Ex 10.2, Ex 10.3, Ex 10.4 |

Number of Questions Solved |
54 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

### Chapter 10 Vector Algebra Exercises 10.1

**Question 1.**

Represent graphically a displacement of 40km, 30° east of north.

**Solution:**

A line segment of 2 cm is drawn on the right of OY making an angle of 30° with it. OP = 40 km,

scale 1cm = 20 km. Vector represents displacement of 40 km 30° east of north.

**Question 2.**

Classify the following measures as scalars and vectors.

(i) 10 kg

(ii) 2 metres north- west

(iii) 40°

(iv) 40 watt

(v) 10^{-19 }coulomb

(vi) 20 m/sec².

**Solution:**

(i) Mass-scalar

(ii) Directed distance-vector

(iii) Temperature-scalar

(iv) Rate of electricity-scalar

(v) Electric charge-vector

(vi) Acceleration-vector

**Question 3.**

Classify the following as scalar and vector quantities

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v)work.

**Solution:**

Scalar Quantity: (i) time period (ii) distance (v) work.

Vector Quantity: (iii) force (iv) velocity

**Question 4.**

In a square, identify the following vectors

(i) Co-initial

(ii) Equal

(iii) collinear but not equal

**Solution:**

(i) Co initial vectors are

(ii) Equal Vectors are

(iii) Collinear but not equal vectors are

**Question 5.**

Answer the following as true or false:

(i) are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

**Solution:**

(i) True

(ii) False

(iii) False

(iv) False.

### Chapter 10 Vector Algebra Exercises 10.2

**Question 1.**

Compute the magnitude of the following vectors:

**Solution:**

**Question 2.**

Write two different vectors having same magnitude.

**Solution:**

Such possible answers are infinite

**Question 3.**

Write two different vectors having same direction.

**Solution:**

Let the two vectors be

Hence vectors have the same direction but different magnitude

**Question 4.**

Find the values of x and y so that the vectors are equal.

**Solution:**

We are given

If vectors are equal, then their respective components are equal. Hence x = 2, y = 3.

**Question 5.**

Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).

**Solution:**

LetA(2, 1) be the initial point and B(-5,7) be the terminal point

∴The vector components are and scalar components are – 7 and 6.

**Question 6.**

Find the sum of three vectors:

**Solution:**

**Question 7.**

Find the unit vector in the direction of the vector

**Solution:**

**Question 8.**

Find the unit vector in the direction of vector , where P and Q are the points (1,2,3) and (4,5,6) respectively.

**Solution:**

The points P and Q are (1, 2, 3) and (4, 5, 6) respectively

**Question 9.**

For given vectors find the unit vector in the direction of the vector

**Solution:**

**Question 10.**

Find a vector in the direction of which has magnitude 8 units.

**Solution:**

The given vector is

**Question 11.**

Show that the vector are collinear.

**Solution:**

vector have the same direction they are collinear.

**Question 12.**

Find the direction cosines of the vector

**Solution:**

let

Now a = 1,b = 2,c = 3

**Question 13.**

Find the direction cosines of the vector joining the points A (1,2, -3) and B(-1, -2,1), directed fromAtoB.

**Solution:**

Vector joining the points A and B is

**Question 14.**

Show that the vector are equally inclined to the axes OX, OY, OZ.

**Solution:**

Let , Direction cosines of vector are

which shows that the vector a is equally inclined to the axes OX, OY, OZ.

**Question 15.**

Find the position vector of a point R which divides the line joining the points whose positive vector are in the ratio 2:1

(i) internally

(ii) externally.

**Solution:**

(i) The point R which divides the line joining the point in the ratio m : n

**Question 16.**

Find position vector of the mid point of the vector joining the points P (2,3,4) and Q (4,1, -2).

**Solution:**

Let

**Question 17.**

Show that the points A, B and C with position vector respectively form the vertices of a right angled triangle.

**Solution:**

**Question 18.**

In triangle ABC (fig.), which of the following is not

(a)

(b)

(c)

(d)

**Solution:**

We know that

Hence option (c) is not correct

**Question 19.**

If are two collinear vectors then which of the following are incorrect:

(a) , for some scalar λ.

(b)

(c) the respective components of are proportional.

(d) both the vectors have same direction, but different magnitudes.

**Solution:**

Options (d) is incorrect since both the vectors , being collinear, are not necessarily in the same direction. They may have opposite directions. Their magnitudes may be different.

### Chapter 10 Vector Algebra Exercises 10.3

**Question 1.**

Find the angle between two vectors with magnitudes √3 and 2 respectively, and such that

**Solution:**

Angle θ between two vectors ,

**Question 2.**

Find the angle between the vectors

**Solution:**

Let

Let θ be the angle between ,

**Question 3.**

Find the projection of the vector , on the line represented by the vector ,

**Solution:**

let

**Question 4.**

Find the projection of the vector on the vector

**Solution:**

let then

**Question 5.**

Show that each of the given three vectors is a unit vector Also show that they are mutually perpendicular to each other.

**Solution:**

**Question 6.**

**Solution:**

Given

**Question 7.**

Evaluate the product :

**Solution:**

**Question 8.**

Find the magnitude of two vectors having the same magnitude and such that the angle between them is 60° and their scalar product is

**Solution:**

We know that

**Question 9.**

Find , if for a unit vector

**Solution:**

Given

**Question 10.**

If such that , then find the value of λ.

**Solution:**

Given

**Question 11.**

Show that for any two non-zero vectors

**Solution:**

are any two non zero vectors

**Question 12.**

If , then what can be concluded about the vector ?

**Solution:**

,

=> = 0

Hence b is any vector.

**Question 13.**

If are the unit vector such that , then find the value of

**Solution:**

We have

**Question 14.**

If either vector then . But the converse need not be true. Justify your answer with an example.

**Solution:**

Given:

To prove:

**Question 15.**

If the vertices A,B,C of a triangle ABC are (1,2,3) (-1,0,0), (0,1,2) respectively, then find ∠ABC.

**Solution:**

Let O be the origin then.

**Question 16.**

Show that the points A (1,2,7), B (2,6,3) and C (3,10, -1) are collinear.

**Solution:**

The position vectors of points A, B, C are

**Question 17.**

Show that the vectors and from the vertices of a right angled triangle.

**Solution:**

The position vectors of the points A, B and C are

and

**Question 18.**

If is a non-zero vector of magnitude ‘a’ and λ is a non- zero scalar, then λ is unit vector if

(a) λ = 1

(b) λ = – 1

(c) a = |λ|

(d) a =

**Solution:**

Given : is a unit vectors

### Chapter 10 Vector Algebra Exercises 10.4

**Question 1.**

Find

**Solution:**

Given

**Question 2.**

Find a unit vector perpendicular to each of the vector , where

**Solution:**

we have

**Question 3.**

If a unit vector makes angle and an acute angle θ with ,then find θ and hence the components of .

**Solution:**

**Question 4.**

Show that

**Solution:**

LHS =

**Question 5.**

Find λ and μ if

**Solution:**

**Question 6.**

Given that . What can you conclude about the vectors ?

**Solution:**

**Question 7.**

Let the vectors are given . Then show that

**Solution:**

Given

are given

**Question 8.**

If either .Is the

converse true? Justify your answer with an example.

**Solution:**

**Question 9.**

Find the area of the triangle with vertices A (1,1,2), B (2,3,5) and C (1,5,5).

**Solution:**

A (1,1,2), B (2,3,5) and C (1,5,5).

**Question 10.**

Find the area of the parallelogram whose adjacent sides are determined by the vectors

**Solution:**

We have

**Question 11.**

Let the vectors such that then is a unit vector if the angle between is

(a)

(b)

(c)

(d)

**Solution:**

Given

**Question 12.**

Area of a rectangles having vertices

(a) sq units

(b) 1sq.units

(c) 2sq.units

(d) 4sq.units

**Solution:**

We hope the NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra help you. If you have any query regarding NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra, drop a comment below and we will get back to you at the earliest.

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