Cross product

The cross product of two 3D vectors is a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span. Unlike the Dot product, the cross product is specific to three dimensions.

Image: Cross product illustration, public domain

Two definitions

Geometric. Let $\theta$ be the angle between $\mathbf{A}$ and $\mathbf{B}$ ($0\le \theta \le \pi$). Then $\mathbf{A}\times \mathbf{B}$ has

• magnitude $|\mathbf{A}\times \mathbf{B}|=|\mathbf{A}||\mathbf{B}|\sin \theta$,
• direction perpendicular to both $\mathbf{A}$ and $\mathbf{B}$, chosen by the right-hand rule: point the fingers of your right hand from $\mathbf{A}$ toward $\mathbf{B}$ through the smaller angle; your thumb points along $\mathbf{A}\times \mathbf{B}$.

The magnitude $|\mathbf{A}||\mathbf{B}|\sin \theta$ is the area of the parallelogram spanned by $\mathbf{A}$ and $\mathbf{B}$ (base $|\mathbf{A}|$, height $|\mathbf{B}|\sin \theta$).

So the cross product encodes both an area and a choice of perpendicular direction — a dual role that becomes central when defining surface normals and the Flux integral.

Algebraic (mnemonic determinant):

$\mathbf{A}\times \mathbf{B}=\left|\begin{array}{ccc}\hat{\mathbf{i}}& \hat{\mathbf{j}}& \hat{\mathbf{k}}\\ a_x& a_y& a_z\\ b_x& b_y& b_z\end{array}\right|=(a_y{b}_z-a_z{b}_y)\hat{\mathbf{i}}-(a_x{b}_z-a_z{b}_x)\hat{\mathbf{j}}+(a_x{b}_y-a_y{b}_x)\hat{\mathbf{k}}.$

Watch the sign on the $\hat{\mathbf{j}}$ term — that’s the standard cofactor-expansion sign.

Properties

• Anticommutative: $\mathbf{A}\times \mathbf{B}=-\mathbf{B}\times \mathbf{A}$. Order matters; flipping reverses the direction.
• Distributive: $\mathbf{A}\times (\mathbf{B}+\mathbf{C})=\mathbf{A}\times \mathbf{B}+\mathbf{A}\times \mathbf{C}$.
• $\mathbf{A}\times \mathbf{A}=0$ (parallelogram with one side has zero area).
• $\mathbf{A}\times \mathbf{B}=0$ iff $\mathbf{A}$ and $\mathbf{B}$ are parallel (or one is zero).

Cyclic unit-vector rules:

$\hat{\mathbf{i}}\times \hat{\mathbf{j}}=\hat{\mathbf{k}},\hat{\mathbf{j}}\times \hat{\mathbf{k}}=\hat{\mathbf{i}},\hat{\mathbf{k}}\times \hat{\mathbf{i}}=\hat{\mathbf{j}}.$

Cyclic order $\hat{\mathbf{i}}\hat{\mathbf{j}}\hat{\mathbf{k}}$ gives the positive sign; reverse order flips the sign ($\hat{\mathbf{j}}\times \hat{\mathbf{i}}=-\hat{\mathbf{k}}$).

Not associative: $\mathbf{A}\times (\mathbf{B}\times \mathbf{C})\ne (\mathbf{A}\times \mathbf{B})\times \mathbf{C}$ in general. The vector triple product has the BAC-CAB identity to handle nested cross products.

Worked example

For $\mathbf{u}=⟨-3,2,1⟩$ and $\mathbf{v}=⟨1,1,0⟩$:

$\mathbf{u}\times \mathbf{v}=\left|\begin{array}{ccc}\hat{\mathbf{i}}& \hat{\mathbf{j}}& \hat{\mathbf{k}}\\ -3& 2& 1\\ 1& 1& 0\end{array}\right|.$

$\hat{\mathbf{i}}$: $2\cdot 0-1\cdot 1=-1$. $\hat{\mathbf{j}}$ (sign flip): $-(-3\cdot 0-1\cdot 1)=1$. $\hat{\mathbf{k}}$: $-3\cdot 1-2\cdot 1=-5$.

So $\mathbf{u}\times \mathbf{v}=⟨-1,1,-5⟩$.

Verify perpendicularity: $\mathbf{u}\cdot (\mathbf{u}\times \mathbf{v})=(-3)(-1)+2\cdot 1+1\cdot (-5)=0$. Same for $\mathbf{v}$.

Applications

Area of a triangle with vertices $P,Q,R$: $\frac{1}{2}|\overrightarrow{PQ}\times \overrightarrow{PR}|$.

Surface normal. Given a parameterization $\mathbf{r}(u,v)$ of a surface, the partial derivatives ${\mathbf{r}}_u,{\mathbf{r}}_v$ are tangent vectors; their cross product ${\mathbf{r}}_u\times {\mathbf{r}}_v$ is normal to the surface, with magnitude equal to the area-element factor. This is the foundation of Flux integral computation.

Torque: $\mathbf{\tau }=\mathbf{r}\times \mathbf{F}$ — perpendicular to both lever arm and applied force, magnitude = $|\mathbf{r}||\mathbf{F}|\sin \theta$.

Angular velocity: $\mathbf{v}=\mathbf{\omega }\times \mathbf{r}$ — the velocity of a rotating point.

Curl. The curl $\nabla \times \mathbf{F}$ is, formally, a “cross product” of the differential operator $\nabla$ with the vector field $\mathbf{F}$.