# Vector Projection

## Key Questions

• A vector is specified by its components along the coordinate axes in a particular coordinate system.
A vector projection of a vector A along some direction is the component of the vector along that direction.
If A makes an angle $\theta$ with the direction in which we are to find it's projection and it's magnitude $A$, the projection is given as $A \cos \theta$.

• Vector projections are used for determining the component of a vector along a direction.

Let us take an example of work done by a force F in displacing a body through a displacement d.
It definitely makes a difference, if F is along d or perpendicular to d (in the latter case, the work done by F is zero).

So, let us for now assume that the force makes an angle $\theta$ with the displacement. In this case the component of force along displacement does all the work.
The component of F along d is $F C o s \theta$ , which is nothing other than the projection of F along d.

Thus, for a general case, work done is given as,

$W = F C o s \theta \cdot d$

Which can be written concisely as,

$W$ = F . d

A vector projection along any direction is the component of a given vector along that direction.

#### Explanation:

If we have to determine the vector projection of vector A with modulus $A$ along a direction with which the vector A makes an angle $\theta$, the projection is given as, $A C o s \theta$

#### Explanation:

The vector projection of $\vec{b}$ onto $\vec{a}$ is

$p r o {j}_{\vec{a}} \vec{b} = \frac{\vec{a} . \vec{b}}{| \vec{a} {|}^{2}} \vec{a}$

Calculate the dot product

$= \vec{a} . \vec{b}$

and calculate the modulus of $\vec{a}$

$= | | \vec{a} | |$