Vector Calculus : Determine a formula for ...

[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? -- I'm thinking there's probably some vector-calculus identities that I should know or something

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? (since it now looks more messy I've probably differentiated wrongly or differentiation at this early step is not correct)

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

 

[royhaas]

You need to know about the scalar triple product.

 

[WalkingInMud]

Thanks heaps

Firstly, we gotta assume that the 3rd derivative exists,

Then, d/dt[r.(r' x r'')] = r'.(r' x r'') + r.[(r'' x r'') + (r' x r''')].

now, r'' x r'' = 0 is a given since they are both parallel (well actually they're the same), also,

r'(r' x r'') = 0 as well -- since (r' x r'') is orthogonal to r' (and r'')

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