Show that Vector function lies on a sphere

[karens]

Homework Statement

Let r1 and r2 be differentiable 3-space vector-valued functions.

Show that for a differentiable 3-space vector-valued function r, the graph of r lies on a sphere centered at the origin if and only if r(t) and r′(t) are orthogonal (perpendicular) for all t.

Homework Equations

Dot products?

The Attempt at a Solution

If R(t) is on the surface of such a sphere then ||R(t)||=C, is constant.

 

[lanedance]

so, if i;m interpreting this correctly if ||r(t)||=c, then you could write it as:

$$\textbf{r}(t) \bullet \textbf{r}(t) = c^2$$
how about differentiating...

note that as the question has "if & only if" you must show both directions, necessary & sufficient" to complete the proof