Can someone help me get started on this question?

Question

I just need to get started, also where ill end up would be nice. I have no idea where to start.

This is the problem:
"Every angle inscribed in a semicircle is a right angle."
Prove this statement using vector method

Im thinking that the cross product will show here because of the right angle part. Just no idea how to get rolling on this, Help would be appreciated. thanks.

Answer 1

It's straight ahead geometry.
Draw a circle with center O. Draw a diameter horizontally through the circle. Label the left intersect as A and the right intersect as B. Now draw a vector from O to a point P on the circle. What tdo you know about triangles AOP and BOP in terms of the angle the vector makes with the diameter? Use law of sines, law of cosines, trig, etc. to show that angle APO and angle BPO add up to 90 degrees and your done.


Doug

Answer 2

I think dot product is a better choice because if you can show that dot product equals zero, you prove the angle between.

Stop here for you since you can get started from here... but let me do it here, ok?








Using the description by doug, vectors AP = OA - OP, PB = OP - OB, and of course, vector OA = -OB. Lets see what is AP.PB?
AP.PB = (OA - OP) . (OP - OB)
= (OA - OP) . (OP + OA)
= (OA - OP) . (OA + OP)
= OA.OA + OA.OP - OP.OA - OP.OP
= r^2 - r^2, where r is the radius of the circle
= 0

Answer 3

there are 4eagles in the circle