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

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

Simple trigonometry gives us:

`cos\ θ = c / | Q |`, so

c= |Q| cosθ

and

`sin\ θ = d / | Q |`, so

d= |Q| sinθ

Since **P** is horizontal,

a= |P|

b= 0

So

ac+bd

= a|Q| cosθ+ 0=

| P ||Q| cosθ

Therefore

P `*` Q=| P ||Q| cosθ=ac+bd

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