Cross Product

$$\vec{a}\times \vec{b}=\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_x& a_y& a_z\\ b_x& b_y& b_z\end{array}\right]=⟨a_y{b}_z-a_z{b}_z,a_z{b}_x-a_x{b}_z,a_x{b}_y-a_y{b}_x⟩$$

• Heuristic for cross product in 3D

$\vec{a}\times \vec{b}=ab\sin \theta \hat{n}$

• $\theta$ is the angle between $\vec{a}$ and $\vec{b}$
• $\hat{n}$ is orthogonal to both vectors and points in the direction of the right thumb when you point the other fingers toward $\vec{a}$ and curl them toward $\vec{b}$

$(\vec{a}\times \vec{b}{)}_i={\epsilon }_{ijk}{a}_j{b}_k$

• ${\epsilon }_{ijk}$ is the Levi-Civita symbol

Properties

$\vec{a}\times (\vec{b}+\vec{c})=(\vec{a}\times \vec{b})+(\vec{a}\times \vec{c})$

• Distributive

$\vec{a}\times \vec{b}=-(\vec{b}\times \vec{a})$

• Anticommutative