d/dt \(\vec u\) = \(\vec w\) x \(\vec u\) and

d/dt \(\vec v\) = \(\vec w\) x \(\vec v\)

Let \(\vec u\) = u₁\(\hat i\) + u₂\(\hat j\) + u₃\(\hat k\) and

\(\vec v\) = v₁\(\hat i\) + v₂\(\hat j\) + v₃\(\hat k\)

\(\vec w\) = w₁\(\hat i\) + w₂\(\hat j\) + w₃\(\hat k\)

d\(\vec u\)/dt = du₁/dt \(\hat i\) + du₂/dt \(\hat j\) + du₃/dt \(\hat k\) = \(\vec w\) x \(\vec u\)

⇒ \(\begin{vmatrix}\hat i&\hat j&\hat k\\w_1&w_2&w_3\\u_1&u_2&u_3\end{vmatrix}\) = du₁/dt \(\hat i\) + du₂/dt \(\hat j\) + du₃/dt \(\hat k\) 

⇒ \(\hat i\)(w₂ u₃ - u₂ w₃) + \(\hat j\)(w₃ u₁ - u₃ w₁) + \(\hat k\)(w₁ u₂ - u₁ w₂)

= du₁/dt \(\hat i\) + du₂/dt \(\hat j\) + du₃/dt \(\hat k\)

⇒ du₁/dt = w₂ u₃ - u₂ w₃,

du₂/dt = w₃ u₁ - u₃ w₁

and du₃/dt = w₁ u₂ - u₁ w₂

Similarly given d\(\vec v\)/dt = \(\vec w\) x \(\vec v\)

So we get dv₁/dt = w₂ v₃ - v₂ w₃,

dv₂/dt = w₃ v₁ - v₃ w₁

and dv₃/dt = w₁ v₂ - v₁ w₂

L.H.S. = d/dt (\(\vec u\) x \(\vec v\)) = d/dt \(\begin{vmatrix}\hat i&\hat j&\hat k\\u_1&u_2&u_3\\v_1&v_2&v_3\end{vmatrix}\)

= d/dt (\(\hat i\)(u₂ v₃ - v₂u₃) + \(\hat j\)(u₃ v₁ - v₃ u₁) + \(\hat k\)(u₁ v₂ - v₁ u₂))

= \(\hat i\)(du₂/dt v₃ + u₂ dv₃/dt - dv₂/dt u₃ - v₂ du₃/dt)

+ \(\hat j\)(du₃/dt v₁ + u₃ dv₁/dt - dv₃/dt u₁ - v₃ du₁/dt)

+ \(\hat k\)(du₁/dt v₂ + u₁ dv₂/dt - dv₁/dt u₂ - v₁ du₂/dt)

= [\(\hat i\)(du₂/dt u₃ - v₂ du₃/dt) + \(\hat j\)(du₃/dt v₁ - v₃ du₁/dt) + \(\hat k\)(du₁/dt v₂ - v₁ du₂/dt)]

+ [\(\hat i\)(u₂ dv₃/dt - dv₂/dt u₃) + \(\hat j\)(u₃ dv₁/dt - dv₃/dt u₁) + \(\hat k\)(u₁ dv₂/dt - dv₁/dt u₂)]

= (d\(\vec u\)/dt x \(\vec v\)) + (\(\vec u\) x d\(\vec v\)/dt)

= (\(\vec w\) x \(\vec u\)) x \(\vec v\)+ (\(\vec u\) x (\(\vec w\) x \(\vec v\)))

= -\(\vec v\)x (\(\vec w\) x \(\vec u\)) + \(\vec u\) x (\(\vec w\) x \(\vec v\))  (∵ \(\hat a\) x \(\hat b\) = -\(\hat b\) x \(\hat a\))

= -((\(\vec v\) . \(\vec u\))\(\vec w\) - (\(\vec v\) . \(\vec w\)) . \(\vec u\)) + (\(\vec u\) . \(\vec v\))\(\vec w\) - (\(\vec u\) . \(\vec w\))\(\vec v\)

(∵ \(\hat a\) x \(\hat b\) x \(\hat c\) = (\(\hat a\) . \(\hat c\))\(\hat b\) - (\(\hat a\) . \(\hat b\)) \(\hat c\))

= -(\(\vec u\) . \(\vec v\)) \(\vec w\) +  (\(\vec v\) . \(\vec w\))\(\vec u\) + (\(\vec u\) . \(\vec v\))\(\vec w\) - (\(\vec u\) . \(\vec w\))\(\vec v\)
(∵ \(\hat a\) . \(\hat b\)  = \(\hat b\) . \(\hat a\))

=  (\(\vec v\) . \(\vec w\))\(\vec u\)) -  (\(\vec u\) . \(\vec w\))\(\vec v\)

=  (\(\vec w\) . \(\vec v\))\(\vec u\) - (\(\vec w\) . \(\vec u\))\(\vec v\) (∵ \(\hat a\) . \(\hat b\) = \(\hat b\) . \(\hat a\))

= \(\vec w\) x (\(\vec u\) x \(\vec v\))

Hence Proved.