Cross Product

The cross product creates a vector perpendicular to two three-dimensional vectors, with magnitude equal to their parallelogram area.

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Three coordinates describe independent movement along three perpendicular axes. A point in space is connected to the coordinate planes by dashed projections. Component arrows build its position one axis at a time.

Unlike the dot product, the cross product returns a direction—one normal to the plane of its two input vectors.

Definition

For

\vec u=\langle u_1,u_2,u_3\rangle,qquad \vec v=\langle v_1,v_2,v_3\rangle,

the cross product is

\vec u\times\vec v =\left\langle u_2v_3-u_3v_2, u_3v_1-u_1v_3, u_1v_2-u_2v_1 \right\rangle.

The output is a three-dimensional vector.

Worked example

Its dot product with the second input is also zero.

Direction and the right-hand rule

The vector $\vec u\times\vec v$ points perpendicular to both inputs. Curl the fingers of the right hand from $\vec u$ toward $\vec v$ through the smaller angle; the thumb gives the cross-product direction.

Reversing order reverses direction:

\vec v\times\vec u=-(\vec u\times\vec v).

The cross product is not commutative.

Finding a plane normal

Two nonparallel direction vectors in a plane have a cross product perpendicular to the plane. This normal vector can produce a scalar plane equation.

If points $A,B,C$ define a plane, compute $\overrightarrow{AB}\times\overrightarrow{AC}$ to obtain a normal.

Common mistakes

Returning a scalar. The cross product returns a vector.

Dropping the negative sign in the middle component. Use the explicit formula.

Assuming order does not matter. Reversing order negates the result.

Using cosine for cross-product magnitude. Area uses sine.

Giving both normal directions as different planes. Opposite normal vectors describe the same plane orientation.

Quick self-check

Is the operation in three dimensions and the output a vector?
Were components and the middle sign computed correctly?
Does the result dot to zero with both inputs?
Does the right-hand rule match the input order?
Is magnitude being interpreted as parallelogram area or twice triangle area?
Are units and geometric meaning stated?