Hello ns1954First, note that the dot product of vectors and is:*****************, where is the angle between the vectors

Next, we'll prove the second result first, because it's more straightforward.

Consider the points on the unit circle . The radii make angles with the positive -axis, measured anticlockwise. So , measured anticlockwise from to .

Using vector notation, we have:Then, using the above result:

***************

, since .

Then the proof of the first result. Note that:, using the result just provedThen, with the point defined as above, consider the point - see the attached diagram. Then the radius makes an angle with the -axis. Therefore .

Then, as before:Grandad