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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!

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    I think I've solved this, I just had a small error manipulating the expressions the first time around. We get: $||a(t_{i}) \times b(t_{i}) - a(t_{i-1}) \times b(t_{i-1})|| \leq ||a(t_i)||||b(t_i)-b(t_{i-1})|| + ||b(t_{i-1})||||a(t_i)-a(t_{i-1})|| \leq A||b(t_i)-b(t_{i-1}|| + B||a(t_i)-a(t_{i-1})||$ (where A and B bound $||a(t)||$ and $||b(t_{i-1})||$) and this lets you bound the sum, I think.2012-06-05

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