Cross Product

Intro

The cross product of two 3D vectors A and B produces a vector C that is perpendicular to both A and B.

$$\mathbf{C}=\mathbf{A}\times \mathbf{B}$$

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Magnitude

The length of C depends on the angle between A and B:

$$|\mathbf{C}|=|\mathbf{A}|,|\mathbf{B}|,\sin \theta$$

where $\theta$ is the angle between A and B.

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Direction (Right-Hand Rule)

• Point your right-hand fingers along A
• Curl them toward B
• Your thumb points in the direction of A × B

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Component Form

If

$$\mathbf{A}=(A_x,A_y,A_z),\mathbf{B}=(B_x,B_y,B_z)$$

then

$$\mathbf{A}\times \mathbf{B}=\left(\begin{array}{c}{A}_y{B}_z-A_z{B}_y{A}_z{B}_x-A_x{B}_z{A}_x{B}_y-A_y{B}_x\end{array}\right)$$

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