Exterior algebra: property of signed area

Question

The wiki on exterior algebras lists of number of properties enjoyed by the signed area, and all of them make sense except for this one:

4. $A(v + rw, w) = A(v, w)$ for any real number $r$, since adding a multiple of $w$ to $v$ affects neither the base nor the height of the parallelogram and consequently preserves its area.

This seems strange since if $v=\mathbf{e_1}$ and $w=\mathbf{e_2}$, then adding a multiple of $w$ to $v$ would change the area, wouldn’t it?

Answer 1

The parallelogram spanned by $\{v,w\}$ has base $||w||$ and height $h = ||v-v_{\perp}||$ , where $v_{\perp} $ is the component of $v$ along $w$. These don't change if we instead consider $\{v + rw,w\}$, and thus the area shouldn't change either.

In your example, the parallelogram spanned by $\{e_1, e_2\}$ is a square that gets deformed into a parallelogram spanned by $\{e_1 + re_2, e_2\}$ with base along $e_2$ and the same height.