Using vector method, prove that the angel in a semi circle is a right angle.

Explanation

We have to prove that the angel in a semi circle is a right angle.
i.e. `angle APB =90^o`
`vec (AB)` is the diameter and
`vec O` is the center
By the law of triangle of vectors:
`vec (PA)= vec(PO) +vec (OA)`
`vec(PB)=vec(PO)+vec(OB)`
`=>vec (PB)=vec (PO) -vec(OA)`
`vec (PA) *vec (PB)=(vec(PO)+vec(OA))*(vec(PO)-vec(OA))`
`=|vec(PO)|^2-|vec(OA)|^2`
`therefore vec(PA)*vec(PB)=0`
so, `vec(PA) bot vec(PB)`
Hence `angle APB =90^o`