Areas in the plane

Areas in the plane: The idea behind cross products is a generalization of the formula for the area of a parallelogram in the plane. So suppose we have two vectors ${\bf a}= \langle a_1, a_2\rangle = a_1 {\bf i} + a_2 {\bf j}$ and ${\bf b}= \langle b_1, b_2\rangle = b_1 {\bf i} + b_2 {\bf j}$ in the plane. We want to compute the area $A({\bf a}, {\bf b})$ of the parallelogram with vertices at the origin, ${\bf a}$, ${\bf b}$ and ${\bf a}+{\bf b}$. This area function should have a few simple properties:

1. It should scale correctly: For any constant $\lambda$, $A(\lambda {\bf a},{\bf b})=A({\bf a}, \lambda {\bf b}) = \lambda A({\bf a},{\bf b})$. This forces us to allow $A({\bf a}, {\bf b})$ to be negative sometimes. If ${\bf b}$ is counter-clockwise of ${\bf a}$, then $A({\bf a}, {\bf b})$ is positive. If ${\bf b}$ is clockwise of ${\bf a}$, then $A({\bf a}, {\bf b})$ is minus the area of the parallelogram.
2. For any vectors ${\bf a}$, ${\bf b}$ and ${\bf c}$, $A({\bf a} + {\bf c}, {\bf b}) = A({\bf a}, {\bf b}) + A({\bf c}, {\bf b})$ and $A({\bf a}, {\bf b}+{\bf c}) = A({\bf a}, {\bf b}) + A({\bf a}, {\bf c})$ . The fancy mathematical term for properties (1) and (2) is that $A({\bf a},{\bf b})$ is a bilinear function of ${\bf a}$ and ${\bf b}$.
3. For any vector ${\bf a}$, $A({\bf a},{\bf a})=0$, since the "parallelogram" is squashed flat.

Notice that

Putting this together, we get the formula for area:

The following video describes determinants of $2\times 2$ and $3 \times 3$ matrices, and explains how they are related to areas and volumes.