Vector Algebra - Notes, Types, Formula, Books, FAQs

Suppose you are playing hide n seek game and one of your friends gives you a hint that he is five steps away from you. Would you be able to catch him??

This will take you many more tries as you are not aware of the direction. Suppose, he is behind you and you start moving in the forward direction. What happens?? Distance will start increasing between both of you. What if he tells you that he is in front of you? Now you can easily catch him as you know the direction as well as the magnitude of the distance of your friend. That's how we make use of vectors algebra in real life unknowingly. There are many more examples like computing the direction of rain, flight of a bird and the list is never-ending.

Vector Algebra is defined as the mathematical operations done on vectors, which is further the foundation of modern-day 3D gaming, animation and widely used in modern physics. This article is about the concept vector algebra class 12. At the JEE level, Vector Algebra is simple to understand. Every year you will get 1 - 2 questions in JEE Main exam as well as in other engineering entrance exams. This chapter helps you in 3-D Geometry and Physics (Kinematics, Work, Energy and Power, electrostatics, etc.). Importance of this chapter can be seen from Physics where you learn why any quantity is scalar or vector? A little mistake in vector algebra costs you negative marks. As compared to other chapters in maths, Vector Algebra requires high accuracy to prepare for the examination. Once you start learning vector algebra you will become familiar with the application of vectors and it helps you to solve the problems based on 3-D geometry and basics used in physics.

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Vector Algebra

Vector- In general terms, a vector is defined as an object having both directions as well as magnitude.

Position vector- Consider a point $P$ in space, having coordinates $(x, y, z)$ with respect to the origin $\mathrm{O}(0,0,0)$. Then, the vector $\overrightarrow{O P}($ or $\vec{r})$ having $O$ and $P$ as its initial and terminal points, respectively, is called the position vector of the point $P$ with respect to $O$ .

Direction Cosines- The position vector of a point $P(x, y, z)$. The angle $\alpha, \beta$, and $Y$ made by the vector with the positive direction of $x, y$, and $z$-axes respectively, are called its directions angles. The cosine values of these angles, i.e., $\cos \alpha, \cos \beta$, and $\cos \gamma$ are called direction cosines of the vector and usually denoted by $l$, $m$, and $n$ respectively.

Types of vector

The types of vector are,

Zero Vector: A vector whose initial and terminal points coincide, is called a zero vector (or null vector). Zero vector can not be assigned a definite direction as it has zero magnitudes. Or, alternatively, otherwise, it may be regarded as having any direction. The vector $\overrightarrow{A A}$ or $\overrightarrow{B B}$ represents the zero vector.

Unit Vector: A vector whose magnitude is unity (i.e., $1$ unit) is called a unit vector. The unit vector in the direction of a given vector $\vec{a}$ is denoted by $\hat{a}$.

Coinitial Vectors: Two or more vectors having the same initial point are called coinitial Vectors.

Collinear Vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

Equal Vectors: Two vectors $\vec{a}$ and $\vec{b}$ are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as $\vec{a}=\vec{b}$.

Negative of a Vector: A vector whose magnitude is the same as that of a given vector (say, $\overrightarrow{A B}$ ), but the direction is opposite to that of it, is called negative of the given vector.

Notes on Vector Algebra

Component Form of Vector

If a vector is represented as $x \hat{i}+y \hat{j}+z \hat{k}$ then it is called component form of the vector. Here, $\hat{i}, \hat{j}$ and $\hat{k}$ representing the unit vectors along the
$x, y$, and $z$-axes, respectively and $(x, y, z)$ represent coordinates of the vector.
Some important points
If $\vec{a}$ and $\vec{b}$ are any two vectors given in the component form $a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ and $b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$, respectively, then,
- The resultant of the vectors is

$
(\vec{a} \pm \vec{b})=\left(a_1 \pm b_1\right) \hat{i}+\left(a_2 \pm b_2\right) \hat{j}+\left(a_3 \pm b_3\right) \hat{k}
$

- The vectors are equal if and only if

$
a_1=b_1, a_2=b_2 \text { and } a_3=b_3
$

- The multiplication of vector $\vec{a}$ by any scalar $\lambda$ is given by

$
\lambda \vec{a}=\lambda a_1 \hat{i}+\lambda a_2 \hat{j}+\lambda a_3 \hat{k}
$

Section Formula

When point $R$ divides $\vec{PQ}$ internally in the ratio of $m:n$ such that $\frac{\overrightarrow{P R}}{\overrightarrow{R Q}}=\frac{m}{n} \quad \vec{r}=\frac{m \vec{b}+n \vec{a}}{m+n}$.
When point $R$ divides $\overrightarrow{P Q}$ externally in the ratio of $m:n$ such that $\frac{\overrightarrow{P R}}{\overrightarrow{Q R}}=\frac{m}{n} \vec{r}_{\text {then }}=\frac{m \vec{b}-n \vec{a}}{m-n}$.

Product of two vector

Multiplication of two vectors is defined in two ways (i) Scalar (or dot) product and (ii) Vector (or cross) product.

In scalar product resultant is scalar quantity.
In vector product resultant is a vector quantity.
Scalar product
Scalar product also known as dot product of two non zero vectors $\vec{a}$ and $\vec{b}$ is denoted by $\vec{a} \cdot \vec{b}$. Scalar product is calculated as $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$ where, $\theta$ is the angle between two non zero given vectors.
Vector product
Vector product of two non zero vectors $\vec{a}$ and $\vec{b}$ is denoted by $\vec{a} \times \vec{b}$. Vector product also known as cross product can be calculated as $\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \hat{n} \quad$ where $\theta$ is the angle between two non zero vectors and $\hat{n}$ is a unit vector perpendicular to both.

List of Topics According to NCERT/JEE MAIN

How to prepare for Vector Algebra?

Vector Algebra is one of the basic topics, you can prepare this topic by understanding a few basic concepts

• Start with the basic concept of vector, understand all the terms used in vector algebra.

• Representation of a vector is an important part of this chapter. It is important for you that you should read all the questions meditatively.

• Vector is all about the direction with magnitude so make sure that the direction given in question and direction obtained in answer match properly.

• Make sure that after studying certain section/concept, solve questions related to those concepts without looking into the solutions and practice MCQ from the above-mentioned books and solve all the previous year problems asked in JEE.

• Don’t let any doubt remain in your mind and clear all the doubts with your teachers or with your friends.

Best Books For Preparation of Vector Algebra

First, finish all the concepts, example and questions given in NCERT Maths Book. You must thorough with the theory of NCERT. Then you can refer to the book Cengage Mathematics Algebra. Vector Algebra is explained very well in this book and there are ample amount of questions with crystal clear concepts. You can also refer to the book Arihant Algebra by SK Goyal or RD Sharma. But again the choice of reference book depends on person to person, find the book that best suits you the best depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.