Q:

Orthogonality on a circle

These theory questions always throw me for a loop.
Ok, the question is:
Suppose that AB is the diameter of the circle with center O and that C is a point on one of the two arcs joining A and B. Show that $\vec{CA}$ and $\vec{CB}$ are orthogonal.
I know I'm suppose to use the dot product somewhere in this, but I don't know how to write this out as a proof.

A:

you have $u \cdot u = v \cdot v = r^2$ and  $$\vec{AC} = \vec{AO} + \vec{OC} = u+v, \quad \vec{BC} = \vec{BO} + \vec{OC} = v - u,   $$ we will take the dot product  $$ \vec{AC}\cdot \vec{BC} = (v+u)\cdot (v-u) =v.v-u.u = r^2 - r^2 = 0. $$  therefore $$AC  \text{ and } BC \text{ are orthogonal. }$$

