Our linear algebra class didn't cover cross products and I require it for a project. I've learned the following methods.
1) $\mathbf{a}\,\times\,\mathbf{b} = ||\mathbf{a}||\,||\mathbf{b}||\,\sin(\theta)\,\mathbf{n}$
2) Determinant method
I don't have a good intuition for why either method works. However, we did learn about orthogonal projections and orthogonal vector spaces so I'm wondering if those methods are in any way related to computing the cross product.
For example, let $\DeclareMathOperator{Span}{Span}W = \Span\{\mathbf{a}, \mathbf{b}\}$. We know that $(Row\, A)^{\perp} = Nul\,A$ so all we need to do is find a vector from $W^{\perp}$ right? (I know that according to methods 1) and 2), the cross product is unique - but does it have to be?)
Anyway, what if we tried finding a vector from $Nul \begin{bmatrix} \mathbf{a}^T \\ \mathbf{b}^T\end{bmatrix}$ where $\mathbf{a}$ and $\mathbf{b}$ are column vectors? In other words, what if we tried finding a vector from the nullspace of a matrix whose rows were the two vectors we are trying to compute the cross product of? Does that say anything about the cross product?