Ethan's Notes

    Cross Product

    #Math

    $$

    \vec{a}\times \vec{b}=\begin{bmatrix}\hat{i} & \hat{j} & \hat{k} \\a_{x} & a_{y} & a_{z} \\b_{x} & b_{y} & b_{z}
    \end{bmatrix}={\left\langle{a_{y}b_{z}-a_{z}b_{z},a_{z}b_{x}-a_{x}b_{z},a_{x}b_{y}-a_{y}b_{x}}\right\rangle}
    $$

    • Heuristic for cross product in 3D

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

    • $\displaystyle \theta$ is the angle between $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$
    • $\displaystyle \hat{n}$ 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 $\vec b$

    $\displaystyle (\vec a\times \vec b)i=\varepsilon{ijk}a_jb_k$

    Properties

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

    • Distributive

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

    • Anticommutative