23. Cross Product Magnitude

The norm of the cross product is related to the angle between the two vectors:

where $𝜃\in \left[0,𝜋\right]$ is the angle between $\overrightarrow{𝑎}$ and $\overrightarrow{𝑏}$.

This is the signed area of the parallelogram spanned by the two vectors:

23.1. Geometric meaning

• $sin\left(𝜃\right)=0$ when $\overrightarrow{𝑎},\overrightarrow{𝑏}$ are parallel (or anti-parallel) — parallelogram degenerates to a line segment, area $0$
• $sin\left(𝜃\right)=1$ when $\overrightarrow{𝑎},\overrightarrow{𝑏}$ are perpendicular — maximum area $\Vert \overrightarrow{𝑎}\Vert \Vert \overrightarrow{𝑏}\Vert$

23.2. Connection to the determinant

The magnitude is the absolute value of the $3\times 3$ determinant with $\overrightarrow{𝑎},\overrightarrow{𝑏}$ as two columns and any orthonormal third column — equivalently the area of the projection of the parallelogram into the plane perpendicular to that third axis.

For a triple of vectors, the triple product $\overrightarrow{𝑎}\cdot \left(\overrightarrow{𝑏}\times \overrightarrow{𝑐}\right)$ gives the signed volume of the parallelepiped — the $3\times 3$ determinant with those vectors as columns.

23.3. Connections

• Cross Product — the operation itself
• Dot vs Cross — $cos$ side-by-side
• Determinant — signed parallelogram / parallelepiped volume
• Angles Between Vectors — recovering $𝜃$ from the formula