Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero

Theorem
Let:
 * $\map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$

be a differentiable vector-valued function.

Let $\map {\mathbf f} x$ be such that its magnitude is constant:
 * $\size {\map {\mathbf f} x} = c$

for some $c \in \R$.

Then the dot product of $\mathbf f$ with its derivative is zero:


 * $\map {\mathbf f} x \cdot \dfrac {\d \map {\mathbf f} x} {\d x} = 0$