Use vectors to show that the angle C inscribed in the adjacent semi-circle is a right angle.?

Question

Ok, so you have a semi circle with origin O. Point C is on the semi circle and so are points A and B forming a triangle whose base is the diameter of the circle. There is a vector a pointing from O to A and is the length of the radius and the vector c pointing from O to C which is also the length of the radius.

I am assuming that I have to show that the dot product between CA and CB = 0

CA can be written as a-c

CB can be written as -a-c

but when i dot them I get -ac+ca but doesnt that give me 2ac and not 0?

Can someone please explain?

Answer 1

oa=-r

ob=r

oc=c1+ic2  (c1^2+c2^2=r^2)      i is perpendicular to aob (say)

ca=-r-c1-ic2

cb=r-c1-ic2

dot pro=-(r-c1)(r+c1)+c2^2

  -(r^2-c1^2)+c2^2

=0

so they r perpendicular

Answer 2

Factor out the c:

-ac+ac=c(a-a)=c*0=0