NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.3 Vector Algebra - PDF Download

1. Product of Two Vectors

Multiplication of two vectors is defined in two ways, namely, scalar (or dot) product where the result is a scalar, and vector (or cross) product where the result is a vector. Based upon these two types of products for vectors, they have found various applications in geometry, mechanics and engineering. Here, you will get to know these two types of products.

• Scalar (or dot) product of two vectors

The scalar product of two given vectors $\vec{a}$ and $\vec{b}$ having angle θ between them is defined as

                              $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$

Also, when $\vec{a} \cdot \vec{b}$ is given, the angle ‘θ’ between the vectors $\vec{a}$ and $\vec{b}$ may be determined by

            $\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

Two important properties of scalar product

Property 1 (Distributivity of scalar product over addition) Let $\vec{a}, \vec{b}$ and $\vec{c}$ be any three vectors, then

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

Property 2 Let $\vec{a}$ and $\vec{b}$ be any two vectors, and l be any scalar. Then

                       $(\lambda \vec{a}) \cdot \vec{b}=(\lambda \vec{a}) \cdot \vec{b}=\lambda(\vec{a} \cdot \vec{b})=\vec{a} \cdot(\lambda \vec{b})$

• Projection of a vector on a line 

A vector $\overrightarrow{\mathrm{AB}}$ makes an angle θ with a given directed line l (say), in the anticlockwise direction. Then the projection of $\overrightarrow{\mathrm{AB}}$ on l is a vector $\vec{p}$ (say) with magnitude $|\overrightarrow{\mathrm{AB}}||\cos \theta|$ , and the direction of $\vec{p}$ being the same (or opposite) to that of the line l, depending upon whether cosθ is positive or negative. The vector $\vec{p}$ is called the projection vector, and its magnitude $|\vec{p}|$ is simply called as the projection of the vector $\overrightarrow{\mathrm{AB}}$ on the directed line l.

Tips for Solving Exercise 10.3 Class 12 Chapter 10 Differential Equations

Ex 10.3 class 12 maths chapter 10 can be solved by following the tips provided by experts of eSaral.

1. Ex 10.3 class 12 maths has two main properties associated with the scalar (or dot) product of two vectors that you need to be well-versed to solve questions of exercise 10.3.

2. You should also go through the projection of a vector on a line to understand the concept properly for solving questions in this exercise.

3. Additionally, you must practice all the examples given before the exe 10.3 for better comprehension of concepts.

Importance of Solving Ex 10.3 Class 12 Maths Chapter 10 Differential Equations

You will gain numerous benefits of solving questions in ex 10.3 class 12 maths chapter 10 Differential Equations. Our subject experts have provided some of them for you which can be checked below.

1. Every topic and concept is deeply explained by experienced teachers of eSaral in NCERT solutions class 12 maths chapter 10 ex 10.3. Therefore, you will feel confident in the examination while solving questions related to this chapter.

2. All the questions of ex 10.3 class 12 maths are elaborated and solved in a step by step manner that will help you to understand the concept behind each question.

3. NCERT solutions for ex 10.3 class 12 maths chapter 10 are also available in free PDF which you can download anytime and solve all the questions error-free. 

4. By solving questions in ex 10.3 class 12 maths, you will be able to improve your mathematical skills that will be beneficial in board exams.

Frequently Asked Questions

Question 1. What are the two main properties of scalar product?

Answer 1. Two main properties of scalar product are as follows.

1. Distributivity of scalar product over addition

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

2. $(\lambda \vec{a}) \cdot \vec{b}=(\lambda \vec{a}) \cdot \vec{b}=\lambda(\vec{a} \cdot \vec{b})=\vec{a} \cdot(\lambda \vec{b})$

Question 2. What are the key topics explained in exercise 10.3 class 12 maths chapter 10?

Answer 2.  The key topics of exercise 10.3 class 12 maths chapter 10 are as follows.

1. Scalar (or dot) product of two vectors

2. Projection of a vector on a line