Cross Product

22. Cross Product

Only defined in $ℝ^{3}$ . Returns a vector orthogonal to the two inputs.

\vec{𝑎} = [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \\ 𝑎_{3} \end{matrix}] \vec{𝑏} = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ 𝑏_{3} \end{matrix}]

\vec{𝑐} = \vec{𝑎} \times \vec{𝑏} = [\begin{matrix} 𝑎_{2} 𝑏_{3} - 𝑎_{3} 𝑏_{2} \\ 𝑎_{3} 𝑏_{1} - 𝑎_{1} 𝑏_{3} \\ 𝑎_{1} 𝑏_{2} - 𝑎_{2} 𝑏_{1} \end{matrix}]

Example

\vec{𝑎} = [\begin{matrix} 1 \\ - 7 \\ 1 \end{matrix}] \vec{𝑏} = [\begin{matrix} 5 \\ 2 \\ 4 \end{matrix}]

\vec{𝑐} = \vec{𝑎} \times \vec{𝑏} = [\begin{matrix} - 7 \cdot 4 - 1 \cdot 2 \\ 1 \cdot 5 - 1 \cdot 4 \\ 1 \cdot 2 - (- 7) \cdot 5 \end{matrix}] = [\begin{matrix} - 30 \\ 1 \\ 37 \end{matrix}]

22.1. Proof: $\vec{𝑐}$ is orthogonal to both $\vec{𝑎}$ and $\vec{𝑏}$

Two vectors are orthogonal when their dot product is zero.

Orthogonal to $\vec{𝑎}$ :

\vec{𝑐} \cdot \vec{𝑎} = [\begin{matrix} 𝑎_{2} 𝑏_{3} - 𝑎_{3} 𝑏_{2} \\ 𝑎_{3} 𝑏_{1} - 𝑎_{1} 𝑏_{3} \\ 𝑎_{1} 𝑏_{2} - 𝑎_{2} 𝑏_{1} \end{matrix}] \cdot [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \\ 𝑎_{3} \end{matrix}]

= 𝑎_{1} 𝑎_{2} 𝑏_{3} - 𝑎_{1} 𝑎_{3} 𝑏_{2} + 𝑎_{2} 𝑎_{3} 𝑏_{1} - 𝑎_{2} 𝑎_{1} 𝑏_{3} + 𝑎_{3} 𝑎_{1} 𝑏_{2} - 𝑎_{3} 𝑎_{2} 𝑏_{1}

= 𝑎_{1} 𝑎_{2} 𝑏_{3} - 𝑎_{1} 𝑎_{3} 𝑏_{2} + 𝑎_{2} 𝑎_{3} 𝑏_{1} - 𝑎_{2} 𝑎_{1} 𝑏_{3} + 𝑎_{3} 𝑎_{1} 𝑏_{2} - 𝑎_{3} 𝑎_{2} 𝑏_{1} = 0

Orthogonal to $\vec{𝑏}$ : same argument with the roles of $\vec{𝑎}, \vec{𝑏}$ swapped.

\vec{𝑐} \cdot \vec{𝑏} = 𝑏_{1} 𝑎_{2} 𝑏_{3} - 𝑏_{1} 𝑎_{3} 𝑏_{2} + 𝑏_{2} 𝑎_{3} 𝑏_{1} - 𝑏_{2} 𝑎_{1} 𝑏_{3} + 𝑏_{3} 𝑎_{1} 𝑏_{2} - 𝑏_{3} 𝑎_{2} 𝑏_{1}

= 𝑏_{1} 𝑎_{2} 𝑏_{3} - 𝑏_{1} 𝑎_{3} 𝑏_{2} + 𝑏_{2} 𝑎_{3} 𝑏_{1} - 𝑏_{2} 𝑎_{1} 𝑏_{3} + 𝑏_{3} 𝑎_{1} 𝑏_{2} - 𝑏_{3} 𝑎_{2} 𝑏_{1} = 0

22.2. See also

Cross Product Magnitude — $‖ \vec{𝑎} \times \vec{𝑏} ‖ = ‖ \vec{𝑎} ‖ ‖ \vec{𝑏} ‖ sin 𝜃$
Dot vs Cross — side-by-side comparison