Question

prove that the angle subtended in half circle is 90 by vector

Answer 1

equation of a circle passing through ends points of diameter is
(x-x1)(x-x2) + (y-y1)(y-y2) = 0

where (x1,y1) and (x2,y2) are ends of diameter.
let they be vector a and b.

take any point P on the circle.

now we need to find angle between PA and PB

PA = a-p and PB = b-p

if their dot product is zero they are perpendicular.

thus PA.PB
= (a-p) . (b-p)
= ((x1-x)i + (y1-x)j) .( (x2-x)i + (y2-x)j )
= (x-x1)(x-x2) + (y-y1)(y-y2)

which equals zero from the circles equation.
Thus they are perpendicular.