How Can We Prove that a Curve with Perpendicular Derivative Lies on a Sphere?

[sharpstones]

My friends and I have been trying to work this one out all night, but to no avail.
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 the origin.

We know the dot product of r(t) and r'(t) = 0 or that r(t) cross r'(t) equals the multiplication of their magnitudes but to go about showing that it is a sphere because of this is causing a great deal of difficulty. Any help would be appreciated

 

[Tide]

If $\vec r \cdot \frac {d \vec r}{dt} = 0$ then $\frac {d}{dt} r^2 = 0$.

 

[HallsofIvy]

Or, to put what Tide said in different words, if the derivative of a vector is always perpendicular to the vector, the vector has constant length.