For convenience, let vectors P and Q be as shown on the graph. (P is horizontal.)

dot product proof

Draw the altitude from vector P to the terminal point (c, d) of Q.

dot product

Simple trigonometry gives us:

`cos\ θ = c / | Q |`, so

c = | Q | cos θ

and

`sin\ θ = d / | Q |`, so

d = | Q | sin θ

Since P is horizontal,

a = | P |

b = 0

So

ac + bd

= a | Q | cos θ + 0

= | P | | Q | cos θ

Therefore

P `*` Q = | P | | Q | cos θ = ac + bd

This result can be generalized for P and Q in any orientation.