NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra

The NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra are given here where the students learn about the difference between a scalar and a vector quantity, their properties, operations of vectors etc. The topic has an important role in helping the students score high marks not only in the board exams but also in the competitive exams. It is important to be prepared for the various problems asked during the 12th board examination. Solving through different exercises and problem sets give students the confidence to write the exams better. These Class 12 NCERT Solutions for Vector Algebra are very easy to understand. Students can also avail these NCERT solutions and download it for free to practice them offline as well.

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NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

Exercise 10.1 Page No: 428

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

Solution:

NCERT Solutions Class 12 Mathematics Chapter 10- image 1

The vector
NCERT Solutions Class 12 MathematicsChapter 10- image 2represents the displacement of 40 km, 30o east of north.

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/s2

Solution:

(i) 10 kg is a scalar quantity because it has only magnitude.

(ii) 2 meters north-west is a vector quantity as it has both magnitude and direction.

(iii) 40° is a scalar quantity as it has only magnitude.

(iv) 40 watts is a scalar quantity as it has only magnitude.

(v) 10–19 coulomb is a scalar quantity as it has only magnitude.

(vi) 20 m/s2 is a vector quantity as it has both magnitude and direction.

3. Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

Solution:

(i) Time period is a scalar quantity as it has only magnitude.

(ii) Distance is a scalar quantity as it has only magnitude.

(iii) Force is a vector quantity as it has both magnitude and direction.

(iv) Velocity is a vector quantity as it has both magnitude as well as direction.

(v) Work done is a scalar quantity as it has only magnitude.

4. In Figure, identify the following vectors.

NCERT Solutions Class 12 Mathematics Chapter 10- image 3

(i) Coinitial (ii) Equal (iii) Collinear but not equal

Solution:

NCERT Solutions Class 12 Mathematics Chapter 10- image 4directions are not the same.

5. Answer the following as true or false.

(i) NCERT Solutions Class 12 Mathematics Chapter 10- image 5 andNCERT Solutions Class 12 Mathematics Chapter 10- image 6are 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.

Vectors
NCERT Solutions Class 12 Mathematics Chapter 10- image 7 and
NCERT Solutions Class 12 Mathematics Chapter 10- image 8are parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

Two vectors having the same magnitude need not necessarily be parallel to the same line.

(iv) False.

Only if the magnitude and direction of two vectors are the same, regardless of the positions of their initial points the two vector are said to be equal.


Exercise 10.2 Page No: 440

1. Compute the magnitude of the following vectors:

NCERT Solutions Class 12 Mathematics Chapter 10- image 9

Solution:

Given vectors are:

NCERT Solutions Class 12 Mathematics Chapter 10- image 10

2. Write two different vectors having same magnitude.

Solution:

NCERT Solutions Class 12 Mathematics Chapter 10- image 11

3. Write two different vectors having same direction.

Solution:

Two different vectors having same directions are:

Let us

NCERT Solutions Class 12 Mathematics Chapter 10- image 12

NCERT Solutions Class 12 Mathematics Chapter 10- image 13

4. Find the values of x and y so that the vectors NCERT Solutions Class 12 Mathematics Chapter 10- image 14are equal

Solution:

Given vectors
NCERT Solutions Class 12 Mathematics Chapter 10- image 15will be equal only if their corresponding components are equal.

Thus, the required values of x and y are 2 and 3 respectively.

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

Solution:

The scalar and vector components are:

The vector with initial point P (2, 1) and terminal point Q (–5, 7) can be shown as,

NCERT Solutions Class 12 Mathematics Chapter 10- image 16

NCERT Solutions Class 12 Mathematics Chapter 10- image 17Thus, the required scalar components are –7 and 6 while the vector components are

NCERT Solutions Class 12 Mathematics Chapter 10- image 186. Find the sum of the vectors

Solution:

Let us find the sum of the vectors:

NCERT Solutions Class 12 Mathematics Chapter 10- image 19

NCERT Solutions Class 12 Mathematics Chapter 10- image 207. Find the unit vector in the direction of the vector

Solution:

We know that

NCERT Solutions Class 12 Mathematics Chapter 10- image 21

NCERT Solutions Class 12 Mathematics Chapter 10- image 22

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:

We know that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 23

9. For given vectors, NCERT Solutions Class 12 Mathematics Chapter 10- image 24and NCERT Solutions Class 12 Mathematics Chapter 10- image 25, find the unit vector in the direction of the vector NCERT Solutions Class 12 Mathematics Chapter 10- image 26

Solution:

We know that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 27

10. Find a vector in the direction of vector NCERT Solutions Class 12 Mathematics Chapter 10- image 28which has magnitude 8 units.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 29

NCERT Solutions Class 12 Mathematics Chapter 10- image 30

11. Show that the vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 31are collinear.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 32

Therefore, we can say that the given vectors are collinear.

12. Find the direction cosines of the vector NCERT Solutions Class 12 Mathematics Chapter 10- image 33

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 34

13. Find the direction cosines of the vector joining the points A (1, 2, –3) and

B (–1, –2, 1) directed from A to B.

Solution:

We know that the

Given points are A (1, 2, –3) and B (–1, –2, 1).

Now,

NCERT Solutions Class 12 Mathematics Chapter 10- image 35

14. Show that the vector NCERT Solutions Class 12 Mathematics Chapter 10- image 36is equally inclined to the axes OX, OY, and OZ.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 37

15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are NCERT Solutions Class 12 Mathematics Chapter 10- image 38 respectively, in the ratio 2:1

(i) internally

(ii) externally

Solution:

We know that

The position vector of point R dividing the line segment joining two points

P and Q in the ratio m: is given by:

NCERT Solutions Class 12 Mathematics Chapter 10- image 39

NCERT Solutions Class 12 Mathematics Chapter 10- image 40

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

Solution:

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

NCERT Solutions Class 12 Mathematics Chapter 10- image 41

17. Show that the points A, B and C with position vectors,NCERT Solutions Class 12 Mathematics Chapter 10- image 42NCERT Solutions Class 12 Mathematics Chapter 10- image 43respectively form the vertices of a right angled triangle.

Solution:

We know

Given position vectors of points A, B, and C are:

NCERT Solutions Class 12 Mathematics Chapter 10- image 44

Hence, proved that the given points form the vertices of a right angled triangle.

18. In triangle ABC (Fig 10.18) which of the following is not true:

NCERT Solutions Class 12 Mathematics Chapter 10- image 45

Solution:

Firstly let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 46

19. If NCERT Solutions Class 12 Mathematics Chapter 10- image 47are two collinear vectors, then which of the following are incorrect:

A. NCERT Solutions Class 12 Mathematics Chapter 10- image 48, for some scalar λ

B. NCERT Solutions Class 12 Mathematics Chapter 10- image 49

C. the respective components of NCERT Solutions Class 12 Mathematics Chapter 10- image 50are proportional

D. both the vectors NCERT Solutions Class 12 Mathematics Chapter 10- image 51have same direction, but different magnitudes

Solution:

We know

NCERT Solutions Class 12 Mathematics Chapter 10- image 52

NCERT Solutions Class 12 Mathematics Chapter 10- image 53


Exercise 10.3 Page No: 447

1. Find the angle between two vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 54andNCERT Solutions Class 12 Mathematics Chapter 10- image 55with magnitudes √3 and 2, respectively havingNCERT Solutions Class 12 Mathematics Chapter 10- image 56.

Solution:

Firstly let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 57

2. Find the angle between the vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 58

Solution:

Let us consider the

NCERT Solutions Class 12 Mathematics Chapter 10- image 59

Hence, the angle between the vectors is cos-1 (5/7).

3. Find the projection of the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 60on the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 61.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 62

4. Find the projection of the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 63on the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 64.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 65

Hence, the projection is 60/114.

5. Show that each of the given three vectors is a unit vector:

NCERT Solutions Class 12 Mathematics Chapter 10- image 66

Also, show that they are mutually perpendicular to each other.

Solution:

It is given that

NCERT Solutions Class 12 Mathematics Chapter 10- image 67

NCERT Solutions Class 12 Mathematics Chapter 10- image 68

NCERT Solutions Class 12 Mathematics Chapter 10- image 696. Find

Solution:

Let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 70

7. Evaluate the productNCERT Solutions Class 12 Mathematics Chapter 10- image 71

Solution:

Let us consider the given expression,

NCERT Solutions Class 12 Mathematics Chapter 10- image 72

8. Find the magnitude of two vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 73, having the same magnitude and such that the angle between them is 60° and their scalar product is ½.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 74

Hence the magnitude of two vectors is 1.

NCERT Solutions Class 12 Mathematics Chapter 10- image 75

Solution:

Let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 76

Hence the value is 13.

10. IfNCERT Solutions Class 12 Mathematics Chapter 10- image 77are such thatNCERT Solutions Class 12 Mathematics Chapter 10- image 78is perpendicular toNCERT Solutions Class 12 Mathematics Chapter 10- image 79, then find the value of λ.

Solution:

We know that the

NCERT Solutions Class 12 Mathematics Chapter 10- image 80

11. Show that NCERT Solutions Class 12 Mathematics Chapter 10- image 81is perpendicular toNCERT Solutions Class 12 Mathematics Chapter 10- image 82, for any two nonzero vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 83.

Solution:

Let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 84

12. IfNCERT Solutions Class 12 Mathematics Chapter 10- image 85, then what can be concluded about the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 86?

Solution:

We know

NCERT Solutions Class 12 Mathematics Chapter 10- image 87

13. If NCERT Solutions Class 12 Mathematics Chapter 10- image 88are unit vectors such that NCERT Solutions Class 12 Mathematics Chapter 10- image 89, find the value of NCERT Solutions Class 12 Mathematics Chapter 10- image 90.

Solution:

Consider the given vectors,

NCERT Solutions Class 12 Mathematics Chapter 10- image 91

Hence the value is -3/2.

14. If either vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 92, thenNCERT Solutions Class 12 Mathematics Chapter 10- image 93. But the converse need not be true. Justify your answer with an example.

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 94

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. [∠ABC is the angle between the vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 95andNCERT Solutions Class 12 Mathematics Chapter 10- image 96]

Solution:

We know

NCERT Solutions Class 12 Mathematics Chapter 10- image 97

Hence, the angle is cos-1 (10/102).

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

Solution:

Let us consider

Given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

Now,

NCERT Solutions Class 12 Mathematics Chapter 10- image 98

Therefore, the given points A, B, and C are collinear.

17. Show that the vectorsNCERT Solutions Class 12 Mathematics Chapter 10- image 99form the vertices of a right angled triangle.

Solution:

Firstly consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 100

NCERT Solutions Class 12 Mathematics Chapter 10- image 101

Solution:

Explanation:

NCERT Solutions Class 12 Mathematics Chapter 10- image 102


Exercise 10.4 Page No: 454

1. FindNCERT Solutions Class 12 Mathematics Chapter 10- image 103, if NCERT Solutions Class 12 Mathematics Chapter 10- image 104andNCERT Solutions Class 12 Mathematics Chapter 10- image 105

Solution:

It is given that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 106

2. Find a unit vector perpendicular to each of the vector NCERT Solutions Class 12 Mathematics Chapter 10- image 107andNCERT Solutions Class 12 Mathematics Chapter 10- image 108, where NCERT Solutions Class 12 Mathematics Chapter 10- image 109andNCERT Solutions Class 12 Mathematics Chapter 10- image 110.

Solution:

It is given that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 111

NCERT Solutions Class 12 Mathematics Chapter 10- image 112

Solution:

Firstly,

NCERT Solutions Class 12 Mathematics Chapter 10- image 113

4. Show that

NCERT Solutions Class 12 Mathematics Chapter 10- image 114

Solution:

Firstly consider the LHS,

We have,

NCERT Solutions Class 12 Mathematics Chapter 10- image 115

5. Find λ and μ if NCERT Solutions Class 12 Mathematics Chapter 10- image 116.

Solution:

It is given that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 117

NCERT Solutions Class 12 Mathematics Chapter 10- image 118

NCERT Solutions Class 12 Mathematics Chapter 10- image 119

NCERT Solutions Class 12 Mathematics Chapter 10- image 120

Solution:

It is given that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 121

NCERT Solutions Class 12 Mathematics Chapter 10- image 122

Solution:

Firstly let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 123

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

Solution:

We know,

NCERT Solutions Class 12 Mathematics Chapter 10- image 124

10. Find the area of the parallelogram whose adjacent sides are determined by the vector NCERT Solutions Class 12 Mathematics Chapter 10- image 125.

Solution:

Let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 126

NCERT Solutions Class 12 Mathematics Chapter 10- image 127

Solution:

Explanation:

NCERT Solutions Class 12 Mathematics Chapter 10- image 128

12. Area of a rectangle having vertices A, B, C, and D with position vectors NCERT Solutions Class 12 Mathematics Chapter 10- image 129and NCERT Solutions Class 12 Mathematics Chapter 10- image 130 respectively is

NCERT Solutions Class 12 Mathematics Chapter 10- image 131

Solution:

Explanation:

NCERT Solutions Class 12 Mathematics Chapter 10- image 132


Miscellaneous Exercise Page No: 458

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Solution:

Let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 133

2. Find the scalar components and magnitude of the vector joining the points P (x1, y1, z1) and Q (x2, y2, z2).

Solution:

Firstly let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 134

3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Solution:

It is given that,

Let O and B be the initial and final positions of the girl respectively.

Then, the girl’s position can be shown as:

NCERT Solutions Class 12 Mathematics Chapter 10- image 135

NCERT Solutions Class 12 Mathematics Chapter 10- image 136

NCERT Solutions Class 12 Mathematics Chapter 10- image 1404. IfNCERT Solutions Class 12 Mathematics Chapter 10- image 137, then is it true thatNCERT Solutions Class 12 Mathematics Chapter 10- image 138? Justify your answer.

Solution:

It is given that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 139

NCERT Solutions Class 12 Mathematics Chapter 10- image 141

5. Find the value of x for whichNCERT Solutions Class 12 Mathematics Chapter 10- image 142is a unit vector.

Solution:

We know,

NCERT Solutions Class 12 Mathematics Chapter 10- image 143

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

NCERT Solutions Class 12 Mathematics Chapter 10- image 144.

Solution:

Let us consider the,

NCERT Solutions Class 12 Mathematics Chapter 10- image 145

7. IfNCERT Solutions Class 12 Mathematics Chapter 10- image 146, find a unit vector parallel to the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 147.

Solution:

Let us consider the given vectors,

NCERT Solutions Class 12 Mathematics Chapter 10- image 148

8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Solution:

Firstly let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 149

NCERT Solutions Class 12 Mathematics Chapter 10- image 150

9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors areNCERT Solutions Class 12 Mathematics Chapter 10- image 151externally in the ratio 1: 2. Also, show that P is the midpoint of the line segment RQ.

Solution:

We know,

NCERT Solutions Class 12 Mathematics Chapter 10- image 152

10. The two adjacent sides of a parallelogram areNCERT Solutions Class 12 Mathematics Chapter 10- image 153and NCERT Solutions Class 12 Mathematics Chapter 10- image 154.

Find the unit vector parallel to its diagonal. Also, find its area.

Solution:

Firstly let us consider,

NCERT Solutions Class 12 Mathematics Chapter 10- image 155

NCERT Solutions Class 12 Mathematics Chapter 10- image 156

11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ areNCERT Solutions Class 12 Mathematics Chapter 10- image 157.

Solution:

Firstly,

Let’s assume a vector to be equally inclined to axes OX, OY, and OZ at angle α.

Then, the direction cosines of the vector are cos α, cos α, and cos α.

Now, we know that

NCERT Solutions Class 12 Mathematics Chapter 10- image 158

Therefore, the direction cosines of the vector which are equally inclined to the axes are

NCERT Solutions Class 12 Mathematics Chapter 10- image 159

Hence proved.

NCERT Solutions Class 12 Mathematics Chapter 10- image 160

Solution:

Assume,

NCERT Solutions Class 12 Mathematics Chapter 10- image 161

NCERT Solutions Class 12 Mathematics Chapter 10- image 162

13. The scalar product of the vectorNCERT Solutions Class 12 Mathematics Chapter 10- image 163with a unit vector along the sum of vectors NCERT Solutions Class 12 Mathematics Chapter 10- image 164and NCERT Solutions Class 12 Mathematics Chapter 10- image 165is equal to one. Find the value ofNCERT Solutions Class 12 Mathematics Chapter 10- image 166.

Solution:

Let’s consider the

NCERT Solutions Class 12 Mathematics Chapter 10- image 167

14. If NCERT Solutions Class 12 Mathematics Chapter 10- image 168are mutually perpendicular vectors of equal magnitudes, show that the vector NCERT Solutions Class 12 Mathematics Chapter 10- image 169is equally inclined to NCERT Solutions Class 12 Mathematics Chapter 10- image 170andNCERT Solutions Class 12 Mathematics Chapter 10- image 171.

Solution:

Lets assume,

NCERT Solutions Class 12 Mathematics Chapter 10- image 172

Hence proved.

15. Prove thatNCERT Solutions Class 12 Mathematics Chapter 10- image 173, if and only if NCERT Solutions Class 12 Mathematics Chapter 10- image 174are perpendicular, givenNCERT Solutions Class 12 Mathematics Chapter 10- image 175.

Solution:

It is given that

NCERT Solutions Class 12 Mathematics Chapter 10- image 176

Hence proved.

NCERT Solutions Class 12 Mathematics Chapter 10- image 177

Solution:

Explanation:

NCERT Solutions Class 12 Mathematics Chapter 10- image 178

NCERT Solutions Class 12 Mathematics Chapter 10- image 179

NCERT Solutions Class 12 Mathematics Chapter 10- image 180

Solution:

Explanation:

NCERT Solutions Class 12 Mathematics Chapter 10- image 181

Hence the correct answer is D.

18. The value of NCERT Solutions Class 12 Mathematics Chapter 10- image 182is

(A) 0 (B) –1 (C) 1 (D) 3

Solution:

Explanation:

It is given that,

NCERT Solutions Class 12 Mathematics Chapter 10- image 183

Hence the correct answer is C.

NCERT Solutions Class 12 Mathematics Chapter 10- image 184

Solution:

Explanation:

NCERT Solutions Class 12 Mathematics Chapter 10- image 185


The major concepts of Maths covered in Chapter 10- Vector Algebra of NCERT Solutions for Class 12 includes: 10.1 Introduction 10.2 Basic Concepts

    • Position Vector
    • Direction Cosines

10.3 Types of Vectors

    • Zero Vector
    • Unit Vector
    • Coinitial Vectors
    • Collinear Vectors
    • Equal Vectors
    • Negative of a Vector

10.4 Addition of Vectors

    • Properties of vector addition

10.5 Multiplication of a Vector by a Scalar 10.5.1 Components of a vector 10.5.2 Vector joining two points 10.5.3 Section formula 10.6 Product of Two Vectors 10.6.1 Scalar (or dot) product of two vectors 10.6.2 Projection of a vector on a line 10.6.3 Vector (or cross) product of two vectors
Exercise 10.1 Solutions 5 Questions
Exercise 10.2 Solutions 19 Questions
Exercise 10.3 Solutions 18 Questions
Exercise 10.4 Solutions 12 Questions
Miscellaneous Exercise On Chapter 10 Solutions 19 Questions

NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra

The chapter Vector Algebra belongs to the unit Vectors and Three – Dimensional Geometry, that adds up to 14 marks of the total marks. There are 4 exercises along with a miscellaneous exercise in this chapter to help students understand the concepts related to Vectors and Vector Algebra clearly. Some of the topics discussed in the tenth Chapter of NCERT Solutions for Class 12 Maths are as follows:

  1. The scalar components of a vector are its direction ratios and represent its projections along the respective axes.
  2. The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as l=(a/r), m=(b/r) n=(c/r)
  3. The vector sum of the three sides of a triangle taken in order is 0.
  4. The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
  5. The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) accordingly as the value of λ is positive (or negative).

The other concepts and topics explained in the chapter can be understood by going through Chapter 10 of the NCERT textbook for Class 12.

Key Features of NCERT Solutions for Class 12 Maths Chapter 10- Vector Algebra

Studying the Vector Algebra of Class 12 enables the students to understand the following: Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, the addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

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