Does anyone see a way of simplifying the following vector expression?

Question

I have the following vector expression: \begin{align} \mathbf{d} = \frac{(\mathbf{c}\!\times\!\mathbf{b})\!\cdot\!(\mathbf{c}\!\times\!\mathbf{b})\big[(\mathbf{a}\!\times\!\mathbf{b})\!\times\!\mathbf{c} + (\mathbf{c}\!\times\!\mathbf{b})\!\times\!\mathbf{a} \big] - (\mathbf{a}\!\times\!\mathbf{b})\!\cdot\!(\mathbf{c}\!\times\!\mathbf{b})\big[(\mathbf{c}\!\times\!\mathbf{b})\!\times\!\mathbf{c}\big]}{\big[(\mathbf{c}\!\times\!\mathbf{b})\!\cdot\!(\mathbf{c}\!\times\!\mathbf{b})\big]^{\frac{3}{2}}} \end{align} Can anyone spot a way of simplifying the above using vector identities?

Answer 1

Since$$(a\times b)\times c=(a\cdot c)b-(b\cdot c)a,$$we have a sum $(a\times b)\times c+(c\times b)\times a=2(a\cdot c)b-(b\cdot c)a-(a\cdot b)c$ and special case $(c\times b)\times c=c^2b-(b\cdot c)c$. And since $(a\times b)\cdot(b\times c)=(a\cdot b)(b\cdot c)-b^2(a\cdot c)$,$$d=\frac{|b\times c|^2[2(a\cdot c)b-(b\cdot c)a-(a\cdot b)c]+[(a\cdot b)(b\cdot c)-b^2(a\cdot c)][c^2b-(b\cdot c)c]}{|b\times c|^3}.$$You can rewrite this a bit further, e.g. with $|b\times c|^2=b^2c^2-(b\cdot c)^2$.