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

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

be a differentiable vector-valued function.

Let $\mathbf f \left({x}\right)$ be such that its magnitude is constant:
 * $\left\lvert{\mathbf f \left({x}\right)}\right\rvert = c$

for some $c \in \R$.

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


 * $\mathbf f \left({x}\right) \cdot \dfrac {\d \mathbf f \left({x}\right)} {\d x} = 0$

Proof
Hence the result.