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Cross Product

Lesson · Intermediate

Geometry: Coordinate Geometry

The cross product of two vectors in R3 produces a vector that is perpendicular to both\text{The cross product of two vectors in }\mathbb{R}^3\text{ produces a vector that is perpendicular to both}

For

u→=(u1,u2,u3),v→=(v1,v2,v3)\overrightarrow{u}=(u_1,u_2,u_3),\qquad \overrightarrow{v}=(v_1,v_2,v_3)

define

u→×v→=(u2v3−u3v2, u3v1−u1v3, u1v2−u2v1)\overrightarrow{u}\times\overrightarrow{v}=(u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1)

A convenient determinant form is

u→×v→=∣ijku1u2u3v1v2v3∣ \overrightarrow{u}\times\overrightarrow{v} = \begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ u_1&u_2&u_3\\ v_1&v_2&v_3 \end{vmatrix}

Geometric Meaning

If θ\theta is the angle between u→\overrightarrow{u} and v→\overrightarrow{v}, then

∣u→×v→∣=∣u→∣∣v→∣sin⁡θ|\overrightarrow{u}\times\overrightarrow{v}|=|\overrightarrow{u}||\overrightarrow{v}|\sin\theta

Right-Hand Rule

To find the direction of A→×B→\overrightarrow{A}\times\overrightarrow{B}, point the fingers of your right hand in the direction of A→\overrightarrow{A} and curl them toward B→\overrightarrow{B}. Your thumb points in the direction of A→×B→\overrightarrow{A}\times\overrightarrow{B}.

Right-hand rule for the cross product showing vectors A and B and the perpendicular vector C equals A cross B
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