This is another one of my “reminder” posts, so look away now if you are familiar with vector dot products.

If we had a two dimensional vector and vector then the dot product would be .

But this can also be expressed as (where is the angle between the two vectors).

Let be the angle makes to the x-axis and the angle made by , then , , and .

Then the dot product can be rewritten as:

=

As we can rewrite this as:

Now and

so we can rewrite the dot product as

As we have .