NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra – PDF Download

Here is a quick review of Class 12 Maths Vector Algebra:

• The position vector of a point P(x, y, z) is given by $\stackrel{\to }{\mathrm{OP}}(\stackrel{\to }{r})=x\hat{i}+y\hat{j}+z\hat{k}$ and its magnitude by $\sqrt{x^2+y^2+z^2}$
• The scalar components of a vector represent its projections along the respective axes, and they are in direction ratios.
• The direction ratios (a, b, c), magnitude (r), and direction cosines (l,m,n) of any vector are related by - $l=\frac{a}{r},m=\frac{b}{r},n=\frac{c}{r}$
• The chapter covers the vector sum of three sides of a triangle, two coinitial vectors, and the multiplication of vector.

To get access to the short revision notes of Chemistry, Physics, and Maths, check - NCERT Class 12 Notes.

Before starting the preparation of any chapter, it is good to know the topics covered in it. See below the topics covered in the Vector Algebra Class 12:

Exercise	Topics Covered
10.1	Introduction
10.2	Some Basic Concepts
10.3	Types of Vectors
10.4	Addition of Vectors
10.5	Multiplication of a Vector by a Scalar
10.6	Product of Two Vectors

Vector Algebra Class 12 Weightage in JEE Main

Exam	Number of Questions	Weightage
JEE Main	2 questions	8%

 

Class 12 Math Chapter 10 Vector Algebra Important Formulae for CBSE and Competitive Exams

Students can check the important topics below;

Basic Vector Operations

• Position Vector: A point $P(x, y, z)$in 3D space has a position vector:

$$\vec{A}=x\hat{i}+y\hat{j}+z\hat{k}$$

• Addition of Two Vectors: If $\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$and $\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$then:

$$\vec{A}+\vec{B}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}$$

• Magnitude of a Vector: For a vector $\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$​ $$|\vec{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$
  
•  Dot Product (Scalar Product): If  $\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$and $\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then;

$$\vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3$$

• Angle between Two Vectors:

$$\vec{A}\cdot \vec{B}=\mathrm{\mid }\vec{A}\mathrm{\mid }\mathrm{\mid }\vec{B}\mathrm{\mid }\cos \theta$$

 

• Cross Product (Vector Product): If $\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$and $\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then;

$$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$

• Magnitude of Cross Product:

$$|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta$$

• Projection of a Vector: 

1. Projection of $\vec{A}$on $\vec{B}$:

   $$\text{Projection}=\frac{\vec{A}\cdot \vec{B}}{\mathrm{\mid }\vec{B}\mathrm{\mid }}$$

2. Vector Component of $\vec{A}$along $\vec{B}$:

   $$\text{Component}=\left(\frac{\vec{A}\cdot \vec{B}}{\mathrm{\mid }\vec{B}{\mathrm{\mid }}^2}\right)\vec{B}$$

• Collinearity of Two Vectors: Two vectors $\vec{A}$ and $\vec{B}$ are collinear if;

$$\vec{A} \times \vec{B} = \vec{0}$$

• Vector Equation of a Line

1. Vector form: A line passing through $\vec{A}$and parallel to $\vec{B}$:

   $$\vec{r}=\vec{A}+t\vec{B},t\in \mathbb{R}$$

2. Cartesian form: A line passing through $(x_1, y_1, z_1)$and parallel to $\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$:

   $$\frac{x-x_1}{b_1}=\frac{y-y_1}{b_2}=\frac{z-z_1}{}$$

Students should download the Vector Algebra Class 12 PDF from the link given below. The solutions are given in a step-by-step format. It is easy to understand and improves the problem-solving skills of students.

Class 12 Math Chapter 10 Vector Solution PDF: Free PDF Download

Related Links

NCERT Notes for Class 11 & 12

Class 12 Maths NCERT Solutions

NCERT Solutions Class 11 and 12

This chapter introduces concepts like vector operations, dot and cross products, scalar triple products, and their real-world significance. Mastering these topics is crucial for excelling in board exams and competitive entrance tests.

 

 

 

Class 12 Vector Algebra Exercise 10.1 Solutions

Class 12 Vector Algebra Exercise 10.1 deals with key concepts such as vectors, their representation, operations like addition, subtraction, and scalar multiplication, as well as their geometric interpretations. Shiksha has provided detailed solutions to all the problems in Vector Algebra Exercise 10.1, ensuring that students grasp the concepts thoroughly. Vector Algebra Class 12 Exercise 10.1 Solutions consists of 5 Questions. Students can check the complete solution for VEctor Algebra Exercise 10.1 below;

Class 12 Vector Algebra Exercise 10.1  Solutions

Q1. Represent graphically a displacement of 40 km, 30° east of north.

A.1.  $40km,30^0$ east of north.

Q2. Check the following measures as scalars and vectors: (i) 10 kg    (ii) 2 meters north-west   (iii) 40° (iv) 40 Watt   (v) 10–19  coulomb   (vi) 20 m/sec2

A.2. (i) 10kg involves only magnitude. So, it is scalar quantity. (ii) 2 meters north-west involves both magnitude and direction. So, it is vector quantity. (iii) 400  involves only magnitude. So, it is scalar quantity. (iv) 400  watts involves only magnitude. So, it is scalar quantity. (v) 10-19 coulomb involves only magnitude. So, it is scalar quantity. (vi) 20m/s-2 involves magnitude and direction. So, it is vector quantity.

Q3. Classify the following as scalar and vector quantities: (i) time period   (ii) distance    (iii) force (iv) velocity   (v) work done

A.3. (i) Time period involves only magnitude. So, it is scalar quantity. (ii) Distance involves only magnitude. So, it is scalar quantity. (iii) Force involves both magnitude and direction. So, it is vector quantity. (iv) Velocity involves both magnitude and direction. So, it is vector quantity. (v) Work done involves only magnitude. So, it is scalar quantity.

Q..4. In the adjoining figure, (a square) identify the following vectors: (i) Coinitial (ii) Equal (iii) Collinear but not equal.

A.4. (a) Vector  $\stackrel{\to }{a}$ and  $\stackrel{\to }{d}$ are co initial same initial point. (b)  $\stackrel{\to }{b}$ and  $\vec{d}$ same magnitude & direction. (c)  $\stackrel{\to }{a}$ and  $\stackrel{\to }{c}$ are collinear but not equal they are parallels their direction are not same.