Dot product and Cross product question

Question

Let $f : \mathbb R → \mathbb R^3$ be a three-times differentiable vector-valued function.

Is this statement true?

$f '(u) \cdot [ f '(u) \times f ''(u)] = 0$

If so how can it be justified or explained? Is it correct to say that these vectors are perpendicular?

Answer 1

$f'(u)\times f''(u)$ is perpendicular to $f'(u)$ (the cross product of vectors is perpendicular to them),

and the dot product of perpendicular vectors is $0$.