Visualizing Unit Vectors

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 \(\overrightarrow{p}\) is given as:

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

Here, \(\hat{p}\) represents a unit vector, \(\overrightarrow{p}\) represents the vector and \(|\overrightarrow{p}|\) represents the magnitude of the vector.

It must be kept in mind that any two unit vectors \(\hat{p}\) and \(\hat{q}\) must not be considered as equal unit vectors just because they have the same magnitude. Since, the direction in which the vectors are taken might be different therefore these unit vectors are different from each other. Therefore, to define a vector both magnitude and direction should be specified.

In Cartesian co-ordinate system, any vector \(\overrightarrow{p}\) can be represented in terms of its unit vectors. The unit vectors in direction of \(x,y\) and \(z\) axes is given by \(\hat{i},\hat{j}\) and \(\hat{k}\) respectively. The position of vector \(\overrightarrow{p}\) can be represented in space with respect to the origin of the given co-ordinate system as:

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

The vector \(\overrightarrow{p}\) can be resolved along the three axes as shown in the given figure. With OM as the diagonal, a parallel piped is constructed whose edges OA,OB and OC lie along the three perpendicular axes.

Unit vectors

From the above figure,

\(\overrightarrow{OA}\) = \(x\hat{i}\)

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

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

The vector \(\overrightarrow{p}\) can be represented as

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

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.’


Practise This Question

Match the following:

(i)Oxygen(A)Metal(ii)Sodium(B) Metalloid(iii)Boron(C)Nonmetal(iv) Gold(v) Carbon dioxide The correct pairs are:-