using vectors show that the angle in a semicircle is a right angle

Hi!

Here is the answer to your question.

Let O (0, 0) be the centre of a semicircle of radius a.

The coordinated of any point P on the circle can be written as P (a cos q, a sin q) where q is the angle formed by OP with positive direction of x-axis.

 

This, means that angle between PA and PB is 90°.

 

Cheers!