Determine a formula for (RE: vector valued functions)

[WalkingInMud]

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?

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

 

[HallsofIvy]

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?

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

What you have now doesn't make sense. r.r' is a number, not a vector. Same thing for r.r". You can't take the cross product. You seem to be assuming that u.(v x w)= (u.v)x(u.w) and, as I just said, that product doesn't make sense.

How about jusst using the product rule:
d/dt[r.(r' x r")]= r'.(r'x r")+ r.(r'+ r")'. The first term is easy: (r' x r") is perpendicular to both r' and r" so its dot product with r' is ?

Now expand (r' x r")' in the same way. Again, the first part of the sum is trivial.