I encoutered following problem when deriving some dynamics equation. Since lack of my background of vector calculus, I am having hard time derivng following equation.
$\nabla_{\vec{x}} \left( (\vec{x}\times \vec{y})^{T} (\vec{x} \times \vec{y}) \right) $
Anyone could help me out?
It may be easier to calculate first the total derivative of $$f(x)=\langle x\times y,x\times y\rangle$$ at $p$ using the product rule: $$D_pf(x)=2\langle x\times y,p\times y\rangle.$$ The using Lagrange we have $$\begin{align} \frac{1}{2}D_pf(x)&=\langle x,p\rangle\langle y,y\rangle -\langle y,p\rangle\langle x,y\rangle\\ &=\langle x\langle y,y\rangle,p\rangle-\langle y\langle x,y\rangle,p\rangle\\ &=\langle x\langle y,y\rangle-y\langle x,y\rangle,p\rangle \end{align}$$ Hence finally $$\nabla f(x)=2(x\langle y,y\rangle-y\langle x,y\rangle)$$ which equals $$2y\times(x\times y)$$ using the Graßmann-identity.