Thread: Vector valued functions : Determine a formula for ...

1. Vector valued functions : Determine a formula for ...

Let r(t) be a v.v.f -with the first and second derivatives r' and r''. Determine formula for
d/dt [r.(r' x r'')] -in terms of r:

How do we approach this one?

maybe:
d/dt [r.(r' x r'')] = d/dt [r.r' x r.r''] = ((r'.r' + r.r'') x r'.r'') + (r.r' x (r'.r'' + r.r''')) ??
...and then what? I probably looking at this completely wrong -- there must be some simple vector calculus identities that make this easy or something.

Is anyone able to give me a starting point -- or starting direction? -- thanks heaps

2. We must assume that r has a third derivative.
$\frac{d}{{dt}}\left[ {r \cdot \left( {r' \times r''} \right)} \right] = r' \cdot \left( {r' \times r''} \right) + r \cdot \left[ {\left( {r'' \times r''} \right) + \left( {r' \times r'''} \right)} \right]$.
But both $r' \cdot \left( {r' \times r''} \right) = 0\quad \& \quad \left( {r'' \times r''} \right) = 0$. Do you see why?
So the final answer would be what?

3. Ahh, ...thanks heaps

r'' x r'' = 0 since they are parallel (the same!), and,

r'.(r' x r '') = 0, since r' x r'' is orthogonal to r' (and r'') ...

so the final answer is r.(r' x r''')