NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.2 Vector Algebra - PDF Download

1. Addition of Vectors

You will learn two laws in addition of vectors.

Triangle law of vector addition - Triangle law of vector addition will teach that the initial point of one vector coincides with the terminal point of the other vector.

This can be expressed as $\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}$

Parallelogram law of vector addition - If we have two vectors represented by the two adjacent sides of a parallelogram in magnitude and direction, then their sum $\vec{a}+\vec{b}$ is represented in magnitude and direction by the diagonal of the parallelogram through their common point. This is known as the parallelogram law of vector addition. 

          $\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{OC}}$

• Properties of vector addition

Property 1: For any two vectors $\vec{a}$ and $\vec{b}$ ,

                            $\vec{a}+\vec{b}=\vec{b}+\vec{a}$

Property 2: For any three vectors a, b and c

                            $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$

 Additive Identity -  For any vector a,

                             $\vec{a}+\overrightarrow{0}=\overrightarrow{0}+\vec{a}=\vec{a}$ 

2. Multiplication of a Vector by a Scalar

Let $\vec{a}$ be a given vector and λ a scalar. Then the product of the vector $\vec{a}$ by the scalar λ, denoted as $\lambda \vec{a}$ , is called the multiplication of vector $\vec{a}$ by the scalar λ.

$\lambda \vec{a}$ is also a vector, collinear to the vector $\vec{a}$ . The vector $\lambda \vec{a}$ has the direction same (or opposite) to that of vector $\vec{a}$ according as the value of λ is positive (or negative). Also, the magnitude of vector $\lambda \vec{a}$  is |λ| times the magnitude of the vector $\vec{a}$ . 

                  $|\lambda \vec{a}|=|\lambda||\vec{a}|$

The vector $-\vec{a}$ is called the negative (or additive inverse) of vector $\vec{a}$ and we always have

            $\vec{a}+(-\vec{a})=(-\vec{a})+\vec{a}=\overrightarrow{0}$

• Components of a vector

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

This form of any vector is called its component form. Where x, y and z are called as the scalar components of $\vec{r}$, and $x \hat{i}, y \hat{j}$ and $z \hat{k}$ are called the vector components of $\vec{r}$ along the respective axes. Sometimes x, y and z are also termed as rectangular components.

• Vector joining two points

The magnitude of vector $\overrightarrow{\mathrm{P}_1 \mathrm{P}_2}$ is given by

$\left|\overrightarrow{\mathrm{P}_1 \mathrm{P}_2}\right|=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$

• Section formula

To find the position of a line segment we use the section formula method that divides a vector in two ways- externally and internally in a given ratio.

Case 1 - When PQ is divided by R internally 

where m and n are positive scalars, we say that the point R divides $\overrightarrow{\mathrm{PQ}}$ internally in the ratio of m : n which is given by

     $\overrightarrow{\mathrm{OR}}=\frac{m \vec{b}+n \vec{a}}{m+n}$

Case 2 - When PQ is divided by R externally

the position vector of the point R which divides the line segment PQ externally in the ratio m : n is given by

        $\overrightarrow{\mathrm{OR}}=\frac{m \vec{b}-n \vec{a}}{m-n}$

Tips for Solving Exercise 10.2 Class 12 Chapter 10 Differential Equations

NCERT solutions class 12 maths chapter 10 ex 10.2 can be solved by following the tips provided by eSaral’s experts of mathematics.

1. This exercise has questions based on some properties that must be practiced before solving questions.

2. There are some important formulas based on the vectors so you need to learn these formulas for better understanding of questions included in ex 10.2.

3. Ex 10.2 class 12 maths solutions have essential concepts that you must study thoroughly for answering the question of exercise and for the questions asked in board exams.
   

Importance of Solving Ex 10.2 Class 12 Maths Chapter 10 Differential Equations

Solving questions in NCERT solutions for class 12 maths chapter 10 ex 10.2 will benefit you in so many ways. Here, we have provided some of the benefits that will help you in board exams.

1. eSaral provides students an in-depth knowledge of topics related to ex 10.2 and more precise solutions for each question so that students can evaluate their preparation and secure high marks in exams.

2. There are many questions in ex 10.2 class 12 maths ch 10 that are based on properties of vector addition, some section formulas that are explained in simple language for better understanding of questions.

3. Practicing questions in NCERT solutions will provide you concept clarity that will help you to solve questions asked in board exams.

4. By revising the concepts and solving questions in NCERT solutions for class 12 maths chapter 10 ex 10.2 will improve your problem solving skills.
   

Frequently Asked Questions

Question 1. Write the section formula when R divides PQ internally?

Answer 1. When R divides PQ internally then the formula is   $\overrightarrow{\mathrm{OR}}=\frac{m \vec{b}+n \vec{a}}{m+n}$

Question 2. Write the two laws of addition of vectors?

Answer 2. The two laws of addition of vectors are as follows.

1. Triangle law of vector addition

2. Parallelogram law of vector addition