Laws of Vector Addition- Parallelogram, Triangle Law, Vector Addition | Class 12 Math Notes Study Material Download Free PDF

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

Laws of Vector Addition

A vector is a physical quantity which is represented both in direction and magnitude. In the upcoming discussion, we shall learn about how to add different vectors. There are different laws of vector addition and they are:

• Triangle law of vector addition
• Parallelogram law of vector addition

Table of Contents

• Laws of Vector Addition
• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition

Triangle Law of Vector Addition

Suppose, we have two vectors A and B as shown.

Now the method to add these is very simple, what we do is to simply place the head of one vector over the tail of the other vector as shown below.

Now join the other endpoints of both the vectors together as shown in the below figure.

The resultant of the given vectors is given by the vector C which represents the sum of vectors  A and B.

i.e. C = A + B

Vector addition is commutative in nature i.e.

if C = A + B; then C = B + A

Or

A + B = C = B + A

Similarly, if you want to subtract both the vectors using the triangle law then simply reverse the direction of any vector and add it to the other one as shown.

Now, this can be represented mathematically as:

   C = A – B

Parallelogram Law of Vector Addition

This law is also very similar to the triangle law of vector addition. Consider the two vectors again.

Now for using the parallelogram law, we represent both the vectors as adjacent sides of a parallelogram and then the diagonal emanating from the common point represents the sum or the resultant of the two vectors and the direction of the diagonal gives the direction of the resultant vector.

The resultant vector is shown by C. This is known as the parallelogram law of vector addition.

By using the orthogonal system of vector representation the sum of two vectors

a =

$\begin{array}{l}{a}_1\hat{i}+a_2\hat{j}+a_3\hat{k}\end{array}$

 and b =

$\begin{array}{l}{b}_1\hat{i}+b_2\hat{j}+b_3\hat{k}\end{array}$

 is given by adding the components of the three axes separately.

i.e. a + b =

$\begin{array}{l}{a}_i\hat{i}+a_2\hat{j}+a_3\hat{k}+b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\end{array}$

$\begin{array}{l}\Rightarrow a+b\end{array}$

=

$\begin{array}{l}(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}\end{array}$

Similarly, the difference can be given as

$\begin{array}{l}a–b\end{array}$

=

$\begin{array}{l}(a_1–b_1)\hat{i}+(a_2–b_2)\hat{j}+(a_3–b_3)\hat{k}\end{array}$

Now let us take an example to understand this topic better.

Example: Let

$\begin{array}{l}\overrightarrow{a}\end{array}$

=

$\begin{array}{l}3\hat{i}+4\hat{j}–7\hat{k}\end{array}$

 and

$\begin{array}{l}\overrightarrow{b}\end{array}$

=

$\begin{array}{l}6\hat{i}+4\hat{j}–6\hat{k}\end{array}$

. Add both the vectors.

Solution: As both the vectors are already expressed in co-ordinate system we can directly add these as follows

$\begin{array}{l}\overrightarrow{a}+\overrightarrow{b}\end{array}$

=

$\begin{array}{l}(3+6)\hat{i}+(4+4)\hat{j}+(-7–6)\hat{k}\end{array}$

or

$\begin{array}{l}\overrightarrow{a}+\overrightarrow{b}\end{array}$

=

$\begin{array}{l}9\hat{i}+8\hat{j}-13\hat{k}\end{array}$

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.

Grab Your Copy Today!

Don’t miss out! Get your copy of Class 12 Math Chapter Vectors by Neeraj Anand and start preparing for success in your board exams. Available now at all leading bookstores and through Anand Technical Publishers.

With Neeraj Anand's expert guidance, you will not only secure excellent marks in vectors but also develop a deeper understanding of how vectors play a role in the world around us.

Let’s make vectors your strength for the CBSE Board Exams! 🌟

$\begin{array}{l}9\hat{i}+8\hat{j}-13\hat{k}\end{array}$

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.

Grab Your Copy Today!

Don’t miss out! Get your copy of Class 12 Math Chapter Vectors by Neeraj Anand and start preparing for success in your board exams. Available now at all leading bookstores and through Anand Technical Publishers.

With Neeraj Anand's expert guidance, you will not only secure excellent marks in vectors but also develop a deeper understanding of how vectors play a role in the world around us.

Let’s make vectors your strength for the CBSE Board Exams! 🌟