Cross Product

An operation called the cross product, allows us to find a vector orthogonal to two given vectors. 

Let $\mathbf{u}=⟨{\mathit{u}}_1,{\mathit{u}}_2,{\mathit{u}}_3⟩$ and $\mathbf{v}=⟨{\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3⟩$. Then, the cross product $\mathbf{u}\times \mathbf{v}$ is vector

$\mathbf{u}\times \mathbf{v}=\left({\mathit{u}}_2{\mathit{v}}_3-{\mathit{u}}_3{\mathit{v}}_2\right)\mathbf{i}-\left({\mathit{u}}_1{\mathit{v}}_3-{\mathit{u}}_3{\mathit{v}}_1\right)\mathbf{j}+\left({\mathit{u}}_1{\mathit{v}}_2-{\mathit{u}}_2{\mathit{v}}_1\right)\mathbf{k}$

$\mathbf{u}\times \mathbf{v}=〈{\mathit{u}}_2{\mathit{v}}_3-{\mathit{u}}_3{\mathit{v}}_2,-\left({\mathit{u}}_1{\mathit{v}}_3-{\mathit{u}}_3{\mathit{v}}_1\right),{\mathit{u}}_1{\mathit{v}}_2-{\mathit{u}}_2{\mathit{v}}_1〉$

The direction of $\mathbf{u}\times \mathbf{v}$ is given by the right-hand rule. If we hold the right hand out with the fingers pointing in the direction of $\mathbf{u}$ then curl the fingers toward vector $\mathbf{v}$, the thumb points in the direction of the cross product, as shown.

The direction of $\mathbf{u}\times \mathbf{v}$ is determined by the right-hand rule.

Notice what this means for the direction of $\mathbf{v}\times \mathbf{u}$. If we apply the right-hand rule to $\mathbf{v}\times \mathbf{u}$, we start with our fingers pointed in the direction of $\mathbf{v}$, then curl our fingers toward the vector $\mathbf{u}$. In this case, the thumb points in the opposite direction of $\mathbf{u}\times \mathbf{v}$.