In physics and mathematics, we generally deal with two kinds of quantities and these quantities are:

- Scalar
- Vector

## Scalar

A scalar is a quantity that has a magnitude only.

e.g. Temperature, Time, Length, Distance, Speed, etc.

## Vector

Quantities that have magnitude and direction both are called vectors.

e.g. velocity, force momentum, torque, electric field, etc.

## Illustration #1: Velocity Vector and Speed Relation

A velocity vector has a magnitude and a direction, its magnitude is called *speed*.

## Notation

A vector ‘A’ is denoted by an arrow sign on the top. (i.e. $\vec{A}$)

Physically, it is denoted by an arrow, where the *tail* of the arrow denotes the initial point and the *tip* denotes the terminal point.

## ——-> A

The length of the arrow tells us about its ‘magnitude’ and the tip of the arrow tells us about its ‘direction’.

## Components of a Vector

A vector $\vec{A}$ in 3 dimensions can be represented with tail at origin ‘O’ of a cartesian coordinate system.

If (A1, A2, A3) is the terminal (ending) point of a vector $\vec{A}$, then it can be written as:

$\vec{A}$ = A1$\hat{i}$ + A2$\hat{j}$ + A3$\hat{k}$

Here, A1$\hat{i}$, A2$\hat{j}$, A3$\hat{k}$ are called *rectangular component vectors *or* components of vector A.*

## Calculating Magnitude of a Vector

Magnitude denotes the length of a vector.

Suppose we have $\vec{A}$ = A1$\hat{i}$ + A2$\hat{j}$ + A3$\hat{k}$,

then, the magnitude of $\vec{A}$ = |$\vec{A}$| = $\sqrt{A1^{2}+A2^{2}+A3^{2}}$

## Illustration #2: Equal Vectors

Vectors A & B are said to be equal only if they have the same magnitude (or length) and the same direction.

## ——-> A

## ——-> B

## Types of Vectors

### 1. Unit Vector

A vector having ‘unit’ magnitude is called a *unit vector*. It is denoted by $\hat{A}$.

$\hat{i}$, $\hat{j}$, $\hat{k}$ are called *rectangular unit vectors*.

Unit vector along the vector $\vec{A}$ = $\frac{\vec{A}}{|\vec{A}|}$

### 2. Polar Vectors

Linear vectors are also called *polar vectors. *These vectors always directed towards some point.

e.g. force acceleration, velocity, etc.

### 3. Axial Vectors

Vectors that are associated with rotation about an axis are known as *axial vectors*.

e.g. torque, angular momentum, etc.

### 4. Null Vectors

Vectors having zero magnitudes are called *null vectors*, i.e. their tail and tip coincide with each other.

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