Scalars And Vectors

# Types of Vectors

## Unit Vector

A vector whose magnitude is unity is called a unit vector. Hence, if $|\overrightarrow{OP}|=1$, then $\overrightarrow{OP}$ is said to be a unit vector. The examples of unit vectors are $(1,0)$ and $(0,1)$.

Unit vector is dimensionless and represents only direction. The unit vector of a non zero vector $\overrightarrow{A}$ in the direction of $\overrightarrow{A}$ is represented by $\hat{A}$ (read A-cap or A-hat) and is given by $\hat{A}=\frac{\overrightarrow{A}}{|\overrightarrow{A}|}$

Hence, if $\overrightarrow{A}=(a_1,a_2)≠0$ then the unit vector $\hat{A}$ in the direction of $\overrightarrow{A}$ is defined by $\left(\frac{a_1}{\sqrt{a_1^2+a_2^2}},\frac{a_2}{\sqrt{a_1^2+a_2^2}}\right)$

The unit vectors along X, Y and Z axes of Cartesian coordinates system are represented by $\hat{i}$, $\hat{j}$ and $\hat{k}$ respectively.

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## Zero Vector (Null Vector)

A vector whose magnitude is zero is called a null vector or zero vector. So, $(0,0)$ is the zero vector and is denoted by $\overrightarrow{O}$. In a zero vector, the origin and the terminal point coincide. The direction of a zero vector is indeterminate.

## Negative Vectors

Two vectors having equal magnitude but opposite directions are called negative vectors. A vector having the same magnitude as that of a given vector $\overrightarrow{OP}$ but direction opposite to it and denoted by $-\overrightarrow{OP}$ or $\overrightarrow{PO}$ is called the negative of $\overrightarrow{OP}$.

## Co-initial Vectors

The vectors starting from a common initial point are said to be co-initial vectors. In the figure given below, $\overrightarrow{A}$ and $\overrightarrow{B}$ start from a same initial point $O$. Hence, $\overrightarrow{A}$ and $\overrightarrow{B}$ are co-initial vectors.

## Collinear Vectors

If two vectors are parallel to each other whatever may be their magnitudes, then they are called collinear vectors. In other words, if angle between two vectors is $0°$ or $180°$, then the vectors are said to be collinear. Collinear vectors are further divided into like and unlike vectors.

### Like Vectors

Two vectors are said to be like if they have the same direction whatever may be their magnitudes.

### Unlike Vectors

Two vectors are said to be unlike if they have opposite directions whatever may be their magnitudes. So, if $\overrightarrow{AB}=-2\overrightarrow{CD}$, then $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are unlike vectors.

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## Orthogonal Vectors

Vectors perpendicular to each other are called orthogonal vectors. In particular case, unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$ along X, Y and Z axes are mutually perpendicular.

## Equal Vectors

Two vectors having equal magnitude and same direction are called equal vectors. If $\overrightarrow{A}$ and $\overrightarrow{B}$ have equal magnitude and same direction. Hence, they are equal vectors; $\overrightarrow{A}=\overrightarrow{B}$.

## Position Vector

The vector representing the position of an object relative to an arbitrary origin is called the position vector. If $O$ is arbitrary origin and $P(x,y,z)$ is the position of a particle, then the position vector of particle $P$ relative to origin $O$ is given by $\overrightarrow{OP}=(x,y,z)$

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## Localised Vector

A vector which passes through a given point and which is parallel to the given vector is said to be a localised vector.

Also, we may write a localised vector $\overrightarrow{AB}$ as an ordered pair $(x,y)$ by accepting the following principle of parallel displacement.

Any vector starting from a point may be displaced to any other point so long as its magnitude and direction remain unaltered.‘ Such an ordered pair $(x,y)$ represents a position vector equivalent to the given localised vector $\overrightarrow{AB}$.