Question

What are vector projections used for?

Answer 1

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 Cos theta` , which is nothing other than the projection of F along d.

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

`W = F Cos theta * d`

Which can be written concisely as,

`W` = F . d