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
.