Edited
Let be the angle between vectors and .
We find the projection of onto by first drawing an altitude (perpendicular to ) from the head of to . We have a right triangle, and by good old SOH CAH TOA, , because is the hypotenuse.
We also have that .
When I substitute that into the equation for the projection of , I get
right?
But that yields a number, not a vector.
What am I (still) doing wrong?