Prove that $$(a \times b)\times(c\times d)$$ is a vector in the direction of the intersection of two planes, one including $a$ and $b$ and the other including $c$ and $d$.
This one is from Hildebrand's Applied Calculus. Can't make it right. I've tried examples with plane equations, but it just gets uglier. Can you help me? Thanks.
You just need to know that $n=a\times b$ is a vector normal to both $a$ and $b$, whence normal to the plane $P(a,b)$ spanned by $a$ and $b$.
So if $m$ is any vector not parallel to $n$ then $r=n\times m$ will be orthogonal to $n$, whence $r$ belongs to the plane $P(a,b)$.
Now use this with $m=c\times d$ and use the same argument for $P(c,d)$.