Can someone help me with the following question. I've been having trouble with problems of this kind for a while now.(adsbygoogle = window.adsbygoogle || []).push({});

Q. If the path u(t) (u is a vector) is differentiable at least three times, simplify:

[tex]

\frac{d}{{dt}}\left[ {\left( {u' \times u''} \right) \bullet \left( {u' - u} \right)} \right]

[/tex]

I can't remember the properties of the dot product but I know that the dot product is an inner product. So I will start by using the distributivity of the inner product.

[tex]\frac{d}{{dt}}\left[ {\left( {u' \times u''} \right) \bullet \left( {u' - u} \right)} \right][/tex]

[tex]

= \frac{d}{{dt}}\left[ {\left( {u' \times u''} \right) \bullet u' - \left( {u' \times u''} \right) \bullet u} \right]

[/tex]

It's not much but it's all I've got at the moment. The cross product will produce a single vector so there probably isn't much that I can do with it. Each of the two expressions inside the square bracket are scalars (scalar functions of t is probably the more accurate description) so I can probably continue as follows.

[tex]

= \frac{d}{{dt}}\left[ {\left( {u' \times u''} \right) \bullet u'} \right] - \frac{d}{{dt}}\left[ {\left( {u' \times u''} \right) \bullet u} \right]

[/tex]

Ok well I'm really stuck here. Does anyone have any suggestions? Thanks.

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# Differentiation and dot product

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