Cross Product Intuition

Just a remark concerning 'problem 2':

The coordinates of the cross product $\bf{a}\times\bf{b}$ are the determinants of the projections of $\bf{a}$ and $\bf{b}$ onto the coordinate planes. So the $x$-coordinate of $\bf{a}\times\bf{b}$ is the area of the parallelogram spanned by the projections of $\bf{a}$ and $\bf{b}$ onto the $yz$-plane. I hope this helps your intuition a bit.

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Geometry

Vector Spaces

Analytic Geometry

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