3D Vectors & the Cross Product

One computation, two payoffs: a perpendicular direction and an area.

The idea

The cross product takes two vectors in space and produces a third vector $\mathbf{u} \times \mathbf{v}$, perpendicular to both. Unlike the dot product, its output is a vector rather than a number, and it exists only in three dimensions. Its definition is a formula in components.

Definition (Cross product).

For $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ in $\mathbb{R}^{3}$, $\mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2,\;\; u_3 v_1 - u_1 v_3,\;\; u_1 v_2 - u_2 v_1).$

Each slot is built from the two coordinates other than its own, multiplied crosswise and subtracted: the first slot uses only the second and third coordinates, and so on around.

Two geometric facts make the formula useful, one about direction and one about length.

Proposition.

Let $\theta$ be the angle between $\mathbf{u}$ and $\mathbf{v}$. Then $\mathbf{u} \times \mathbf{v}$ is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, and $|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}|\,|\mathbf{v}|\sin\theta,$ the area of the parallelogram with sides $\mathbf{u}$ and $\mathbf{v}$.

Of the two directions perpendicular to the plane of $\mathbf{u}$ and $\mathbf{v}$, the right-hand rule picks which one — point the fingers of your right hand along $\mathbf{u}$, curl them toward $\mathbf{v}$, and your thumb points along $\mathbf{u} \times \mathbf{v}$. Parallel vectors give $\theta = 0$, so their cross product is the zero vector — they span no parallelogram.

Ways to work on it

• Walkthrough. The component formula, perpendicularity, anti-commutativity, and area.
• Practice. Compute components of u × v.
• Hardest. Parallelogram area via the cross product's length.

Not sure where to start? Take the ten-question placement test.