Let r(t) be a differentiable vector valued function of 1 variable over the real line(so we can take the derivative of each component).

Question

Show that if r(t)*r'(t)=0 for every t, then

|r(t)| = |r(0)| for every t

That is, |r(t)| must be a constant function.

Note that this means r(t) must be a curve that lies on a sphere center at the origin

Answer 1

Let r(t) be an n-dimensional vector (r1(t), ..., rn(t)).

For convenience I will drop the variable (t) and write r = (..., ri, ...).

Then

|r|^2 = SUM(ri^2)

(|r|^2)' = SUM(2*ri*ri') = 2*r*r' = 0

Therefore |r|^2 is constant.

Answer 2

I assume your question is if for every t, then .

Now we have:

.

since , then we have , which means is constant.

Based on the definition of dot product, , therefore, is constant and we have

.