I don't understand the motivation behind defining cross products the way they're defined.
Given two vectors $\vec{A}$ and $\vec{B}$ in $\mathbb{R^3}$, I can find a third vector $\vec{C}$ such that $\vec{C}$ is normal to $\vec{A}$ and $\vec{B}$ by using the following formula:
$$\vec{C} = r \left\langle \;1,\;\frac{a_xb_z-b_xa_z}{b_ya_z-a_yb_z}, \;\frac{a_x+\frac{a_xb_z-b_xa_z}{b_ya_z-a_yb_z}a_y}{a_z}\;\right\rangle \quad\text{where}\quad r \in \mathbb{R}$$
EDIT: all entries are non-zero
Best Answer
Yes, $v\times w$ is orthogonal to both $v$ and $w$. But it also has the property that $\lVert v\times w\rVert=\lVert v\rVert.\lVert w\rVert.\sin\theta$, where $\theta$ is the angle between $v$ and $w$. In particular, it provides an easy way to find an unit vector which is orthogonal to two given unit vectors which are already orthogonal to each other.