# What are vector projections used for?

May 31, 2015

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