VGUH EPM · Mathematics

Why two vectors multiply into an area

Take $\vec{a}$ as the base, with length $|\vec{a}|$. The parallelogram's area is base times height, but the height is only the part of $\vec{b}$ that stands perpendicular to the base, not its full length. Lean $\vec{b}$ toward $\vec{a}$ and the shape flattens: the area falls even though neither length changed.

That perpendicular height equals $|\vec{b}|\sin \theta$, where $\theta$ is the angle between the vectors, so the area is base times height:

The cross product $|\vec{a}\times \vec{b}|$ is defined as exactly that quantity, so it is not new magic: it is base times height in one stroke, with $\sin \theta$ picking out the perpendicular part. Line the vectors up ($\theta =0$) and it drops to 0; square them off ($\theta =90^{\circ }$) and it reaches its maximum $|\vec{a}||\vec{b}|$.

A triangle on the same two vectors is half of that. In space $\vec{a}\times \vec{b}$ turns into a vector perpendicular to both, yet its length is still that area, so for three points you build two edge vectors and halve the length of their cross product:

Drag the two vectors to feel how the area follows their cross product.