Calculus Proof

[sam101]

Do any of you know how to do this problem?

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.

My teacher said that the best way to approach this problem was to take the derivative of (abs r(t))^2, but I do not know what he means. Could somebody please help me with this problem. It is due very soon.

 

[pka]

\(\displaystyle \frac{{d\left[ {r \cdot r} \right]}}{{dt}} = 2r \cdot r' = 0\; \Rightarrow \;\left\|r\right\| = c\)
Because the derivative is zero the function is constant( has the same length everwhere).