Vector function, position and tangent vectors

[Maianbarian]

Homework Statement

If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t) show that the curve lies on a sphere with center at the origin

Homework Equations

The Attempt at a Solution

I have no idea how to even approach this problem, if somebody could give me a nudge in the right direction it would be much appreciated.

 

[Dick]

Differentiate r(t).r(t) (the dot product).

 

[Maianbarian]

Ah, I get it now.

We know that r(t).r(t)=|r(t)|^2. Differentiating the dot product we get:

d/dt [r(t).r(t)]= 2r'(t).r(t) = 0 (as r'(t) is orthogonal to r(t))

thus |r(t)|^2 is constant, and so |r(t)| is constant, which corresponds to a sphere with the centre at the origin.

Thanks Dick :)