Cross product

The cross product (or vector product) between two vectors produces a vector that is orthogonal to both input vectors, i.e., for some vector $r_1$, $r_2$:

$$r_1\times r_2=r_3$$ $$r_3\cdot r_1=0;r_3\cdot r_2=0$$

We can also relate the angle between two vectors to the cross product, where $0\le \theta \le \pi$.

$$\Vert \vec{v}\times \vec{w}\Vert =\Vert \vec{v}\Vert \Vert \vec{w}\Vert \sin \theta$$

The key implication of this is that the magnitude of the cross product is the area of the parallelogram created by $\vec{v}$ and $\vec{w}$.

Computations

Determinants are also very useful tools for computing the cross product by hand in ${\mathbb{R}}^3$. We can do Laplace expansion where the first row are the unit vectors $\hat{i},\hat{j},\hat{k}$, and the corresponding next two rows are the two vectors. In MATLAB, we can use cross(v, w).

$$\vec{w}=\vec{u}\times \vec{v}=\det \left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ u_x& u_y& u_z\\ v_x& v_y& v_z\end{array}\right]$$

There’s also a useful trick for computing the cross product that holds in all three major coordinate systems in ${\mathbb{R}}^3$:

• Write out the unit vectors in order twice.
  • For rectangular, this is $\hat{i},\hat{j},\hat{k},\hat{i},\hat{j},\hat{k}$.
  • For polar, this is $\hat{r},\hat{\phi },\hat{z},\hat{r},\hat{\phi },\hat{z}$.
  • For spherical, this is $\hat{R},\hat{\Theta },\hat{\phi },\hat{R},\hat{\Theta },\hat{\phi }$.
• Then, when we want to compute a cross product, we take the next unit vector as the result. If we move right, it’s positive. If we move left, it’s negative.
  • For example, $\hat{i}\times \hat{j}=\hat{k}$. And $\hat{k}\times \hat{j}=\hat{i}$.

Properties

$$\vec{v}\times \vec{v}=\vec{0}$$

The cross product is non-commutative:

$$\vec{v}\times \vec{w}\ne \vec{w}\times \vec{v}$$ $$\vec{v}\times \vec{w}=-(\vec{w}\times \vec{v})$$

If and only if $\vec{v}$ is parallel to $\vec{w}$:

$$\vec{v}\times \vec{w}=\vec{0}$$

In general, the cross product fails to be associative:

$$\vec{u}\times (\vec{v}\times \vec{w})\ne (\vec{u}\times \vec{v})\times \vec{w}$$

See also

• Dot product