Unit Vectors:

The physical quantities for which both magnitude and direction are defined distinctly are known as vector quantities.The vectors are denoted by putting an arrow over the denotations representing them.

For Example: To define acceleration of a vehicle, along with its magnitude, its direction must also be specified. It can be represented in vector form as \(\overrightarrow{a}~m/s^2\)

Vectors can be easily represented using the co-ordinate system in three dimensions. The vectors having magnitude of one unit are known as unit vectors. A vector can be represented in space using unit vectors. Sometimes unit vector is also known as a direction vector. A unit vector is represented using a lowercase letter with a cap (‘^’) symbol along with it.

A unit vector p ̂having the same direction as vector \(\overrightarrow{p}\)

\(\hat{p}~=~\frac{\overrightarrow{p}}{|\overrightarrow{p}|}\)

Here, \(\hat{p}\)

It must be kept in mind that any two unit vectors \(\hat{p}\)

In Cartesian co-ordinate system, any vector \(\overrightarrow{p}\)

\(\overrightarrow{p}~=~x\hat{i}~+~y\hat{j}~+~z\hat{k}\)

The vector \(\overrightarrow{p}\)

From the above figure,

\(\overrightarrow{OA}\)

\(\overrightarrow{OB}~~=~~y\hat{j}\)

\(\overrightarrow{OC}~~=~~z\hat{k}\)

The vector \(\overrightarrow{p}\)

\(\overrightarrow{p}~=~\overrightarrow{OM}~=~x\hat{i}~+~y\hat{j}~+~z\hat{k}\)

This is known as the component form of a vector. This represents the position of given vectors in terms of the three co-ordinate axes.’