Differentiation of vectors: prove d(v x (r x v)) / dt = ...

[jacksonjoh]

(b) If r, v, and a are the position vector, velocity, and acceleration of a particle at time t, respectively, prove that:

. . . . .\(\displaystyle \dfrac{d}{dt}\, \left(v\, \times\, (r\, \times\, v)\right)\, =\, 2\, (v\, \cdot\, a)\, r\, -\, (r\, \cdot\, a)\, v\, -\, (v\, \cdot\, r)\, a\)

 

[julianabrito12]

It is know that

a x (b x c) = b(a . c) - c (a . b)

so v x (r x v) becomes r(v . v) - v(r . v)

now we apply the product rule for integrals with dr/dt = v, dv/dt = a

d/dt [r(v . v) - v(r . v)] = dr/dt . (v . v) + r 2 v . dv/dt - dv/dt . r v - v . dr/dt . v - v . r dv/dt =

= v . (v . v) + 2 (v . a) r - (r . a) v - (v . v) . v - (v . r)a = 2 (v . a) r - (r . a) v - (v . r)a

qed.