Scalar or Dot Product Of Two Vectors | Projection of Vector | Definition, Properties, Formulas and Examples | Class 12 Math Notes Study Material Download Free PDF

Vector is a quantity that has both magnitude and direction. Some mathematical operations can be performed on vectors such as addition and multiplication. The multiplication of vectors can be done in two ways, i.e. dot product and cross product. In this article, you will learn the dot product of two vectors with the help of examples.

Class 12 Math Chapter Vectors: Important Topics for CBSE Board Exam Students
Written by Author Neeraj Anand (Director of Anand Classes Jalandhar)
Published by Anand Technical Publishers

The definition of dot product can be given in two ways, i.e. algebraically and geometrically. Algebraically, the dot product is defined as the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the two vectors’ Euclidean magnitudes and the cosine of the angle between them. Both the definitions are equivalent when working with Cartesian coordinates. However, the dot product of two vectors is the product of the magnitude of the two vectors and the cos of the angle between them. To recall, vectors are multiplied using two methods

• scalar product of vectors or dot product
• vector product of vectors or cross product

The difference between both the methods is just that, using the first method, we get a scalar value as resultant and using the second technique the value obtained is again a vector in nature.

Table of Contents

• Dot Product of Vectors
• Projection of Vectors
• Dot Product Properties of Vector
• Dot Product of Two Vectors Example Questions

Dot Product of Vectors

The scalar product of two vectors a and b of magnitude |a| and |b| is given as |a||b| cos θ, where θ represents the angle between the vectors a and b taken in the direction of the vectors.

We can express the scalar product as:

a.b=|a||b| cosθ

where |a| and |b| represent the magnitude of the vectors a and b while cos θ denotes the cosine of the angle between both the vectors and a.b indicate the dot product of the two vectors.

In the case, where any of the vectors is zero, the angle θ is not defined and in such a scenario a.b is given as zero.

Projection of Vectors

BP is known to be the projection of a vector a on vector b in the direction of vector b given by |a| cos θ.

Similarly, the projection of vector b on a vector a in the direction of the vector a is given by |b| cos θ.

Projection of vector a in direction of vector b is expressed as

$\begin{array}{l}BP=\frac{a.b}{|b|}\end{array}$

$\begin{array}{l}\Rightarrow \overrightarrow{BP}=\frac{a.b}{|b|}\times \hat{b}\end{array}$

$\begin{array}{l}\Rightarrow \overrightarrow{BP}=\frac{a.b}{|b|}.\frac{b}{|b|}\end{array}$

$\begin{array}{l}\Rightarrow \overrightarrow{BP}=\frac{a.b}{|b{|}^2}b\end{array}$

Similarly, projection of vector b in direction of vector a is expressed as

$\begin{array}{l}BQ=\frac{a.b}{|a|}\end{array}$

$\begin{array}{l}\Rightarrow \overrightarrow{BQ}=\frac{a.b}{|a|}\times \hat{a}\end{array}$

$\begin{array}{l}\Rightarrow \overrightarrow{BQ}=\frac{a.b}{|a|}\frac{a}{|a|}\end{array}$

$\begin{array}{l}\Rightarrow \overrightarrow{BQ}=\frac{a.b}{|a{|}^2}a\end{array}$

Thus, we see that the dot product of two vectors is the product of magnitude of one vector with the resolved component of the other in the direction of the first vector.

Dot Product Properties of Vector

• Property 1: Dot product of two vectors is commutative i.e. a.b = b.a = ab cos θ.
• Property 2: If a.b = 0 then it can be clearly seen that either b or a is zero or cos θ = 0. $\begin{array}{l}\Rightarrow \theta =\frac{\pi }{2}.\end{array}$  It suggests that either of the vectors is zero or they are perpendicular to each other.
• Property 3: Also we know that using scalar product of vectors (pa).(qb)=(pb).(qa)=pq a.b
• Property 4: The dot product of a vector to itself is the magnitude squared of the vector i.e. a.a = a.a  cos 0 = a2
• Property 5: The dot product follows the distributive law also i.e. a.(b + c) = a.b + a.c
• Property 6: In terms of orthogonal coordinates for mutually perpendicular vectors it is seen that $\begin{array}{l}\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1\end{array}$
• Property 7: In terms of unit vectors, if $\begin{array}{l}a=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}andb=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\end{array}$  then $\begin{array}{l}a.b=(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}).(b_1\hat{i}+b_2\hat{j}+b_3\hat{k})\end{array}$

$\begin{array}{l}\Rightarrow a_1{b}_1+a_2{b}_2+a_3{b}_3=abcos\theta \end{array}$

Dot Product of Two Vectors Example Questions

Example 1: Let there be two vectors [6, 2, -1] and [5, -8, 2]. Find the dot product of the vectors.

Solution:

Given vectors: [6, 2, -1] and [5, -8, 2] be a and b respectively.

a.b = (6)(5) + (2)(-8) + (-1)(2)

a.b = 30 – 16 – 2

a.b = 12

Example 2: Let there be two vectors |a|=4 and |b|=2 and θ = 60°. Find their dot product.

Solution:

a.b = |a||b|cos θ

a.b = 4.2 cos 60°

a.b = 4.2 × (1/2)

a.b = 4

Categories

For CBSE Class 12 students, Vectors is one of the most significant chapters that can shape your overall performance in the board exams. Not only is it important for mathematics but also plays a crucial role in subjects like physics and engineering. To help you master this chapter, Neeraj Anand, Director of Anand Classes, Jalandhar, has meticulously crafted a guide to understand and excel in vectors. The book, published by Anand Technical Publishers, is designed to make complex vector concepts accessible and easy to grasp.

Key Topics in Class 12 Math Vectors for CBSE:

1. Unit Vector: Learn how to find vectors of unit length and their role in determining direction in vector equations.
2. Zero (Null) Vector: Understand the concept of a vector with zero magnitude and its properties.
3. Magnitude of a Vector: Master how to calculate the length or magnitude of a vector in different coordinate systems.
4. Addition and Subtraction of Vectors: Understand how to add and subtract vectors algebraically and geometrically.
5. Scalar Multiplication: Learn how multiplying a vector by a scalar changes its magnitude and direction.
6. Dot Product (Scalar Product): Get a solid grasp of dot product and its geometric interpretation for calculating angles between vectors.
7. Cross Product (Vector Product): Learn the properties of the cross product, including its application in finding perpendicular vectors and areas of parallelograms.

Why This Book is a Must-Have:

• Clear and Concise Explanation: Neeraj Anand’s teaching approach makes complex topics easy to understand with detailed explanations and step-by-step solutions.
• Variety of Practice Problems: With ample examples and practice questions, students can build confidence and mastery over the subject.
• Student-Centered Approach: The book is designed with the CBSE board exam syllabus in mind, ensuring that students are fully prepared for all possible questions.
• Application in Real Life: The author also explains how vector concepts are used in physics and engineering, adding value beyond just academic performance.

Perfect for CBSE Board Exam Preparation:

If you’re looking for a resource to help you ace the Vectors chapter in your Class 12 CBSE board exams, this book is your go-to guide. Whether you need to understand basic concepts or solve advanced problems, Neeraj Anand’s guide will provide you with the clarity and practice you need.

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