I'd like some help solving this exercise:
Let $a(t), b(t) : [c,d] \rightarrow R^3$ be different paths. If $a(t)$ and $b(t)$ are paths with finite length, does the cross product of $a(t)$ and $b(t)$ have length?
So far I've tried somehow using the formula $||a\times b||=||a||||b||sin(\theta)$ in the definition of path length as a supremum, but I haven't managed to bound this expression from above.
Any help would be appreciated. Thanks!