Cross Product

Definition

$$\begin{gathered} \vec{a}\times \vec{b} \\ =\det \left(\begin{array}{ccc}{\hat{e}}_1& {\hat{e}}_2& {\hat{e}}_3\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right) \\ =⟨a_2{b}_3-b_2{a}_3,-(a_1{b}_3-b_1{a}_3),a_1{b}_2-a_2{b}_1⟩ \end{gathered}$$

Involves finding the Determinant of a matrix

Geometric Properties

• $\vec{a}\perp \left(\vec{a}\times \vec{b}\right)$
• $\Vert \vec{a}\times \vec{b}\Vert =\Vert \vec{a}\Vert \Vert \vec{b}\Vert \sin \theta$ (area of a parallelogram with sides of $\vec{a}$ and $\vec{b}$)
• Direction by right hand rule
  1. Place hand along $\vec{a}$
  2. Fold fingers in the direction of the smallest angle between the vectors
  3. Thumb is pointing in the direction of the cross product

Applications

• $\vec{a}\times \vec{b}=\vec{0}⟺\vec{a}=s\vec{b}⟺\vec{a}\parallel \vec{b}$
• $\Vert \vec{a}\times \vec{b}\Vert$ is the area of a parallelogram with sides of $\vec{a}$ and $\vec{b}$
  • Can half it to find the area of a triangle given two sides as vectors
• Torque: $\vec{T}=\vec{r}\times \vec{F}$
  • No rotational acceleration if ${\vec{T}}_{net}=\sum _n{\vec{r}}_n\times {\vec{F}}_n=\vec{0}$

Algebraic Manipulation

• $\vec{a}\times \vec{b}=-\vec{b}\times \vec{a}$ (skew-symmetry/anti-commutativity)
• $(\vec{a}+\vec{b})\times \vec{c}=\vec{a}\times \vec{c}+\vec{b}\times \vec{c}$
• $(s\vec{b})\times \vec{b}=0$

The Double Cross Product

• $\vec{a}\times (\vec{b}\times \vec{c})=\vec{b}(\vec{a}\cdot \vec{c})-\vec{c}(\vec{a}\cdot \vec{b})$
  • The ”bac-cab” rule
• Jacobi Identity
  • $\vec{a}\times (\vec{b}\times \vec{c})+\vec{b}\times (\vec{c}\times \vec{a})+\vec{c}\times (\vec{b}\times \vec{c})=\vec{0}$
  • All cyclic permutations
  • Application of the ”bac-cab” rule