Geometrically, the cross product of two vectors produces a three-dimensional vector that is orthogonal (perpendicular) to the input vectors. The magnitude of the resulting vector is equal to the area formed between the two vectors. For example, the cross product of the two vectors below produces as a vector as a result.

The fact that the resulting vector is orthogonal is demonstrated in this example because the two input vectors lie in the coordinate plane. A perpendicular vector then has to have zero for the and components. The two input vectors are illustrated below.

The fact that the magnitude of the resulting vector is equal to the area of the parallelogram can be seen by calculating the area between the two vectors in the plane.

The magnitude of the vector is given below. This can be calculating using this formula.

The area of the parallelogram is below as the area of the rectangle minus the area of the four triangles.

There are two formulas to compute the cross-product of two vectors. The first formula calculates the cross-product using the determinant. The second formula calculates the magnitude of the cross product, which is also equal to the parallelogram area between the two input vectors.

The cross-product operator is given by the formula shown above. This formula calculates the , and components of the resulting vector. For example, to calculate the cross product of the two vectors below start by setting up the formula.

Set up the formula.

Substitute the components of the vectors into the formula.

Apply the 2D determinant formula, then simplify the arithmetic.

Change to the vertical notation if desired.

The second formula related to the cross product calculates the magnitude of the resulting vector which also happens to be equal to the area between the two input vectors. This formula is given by the magnitude of the two vectors multiplied by the angle between them.

For example, to calculate

Set up the formula.

Apply the magnitude of vector formula.

The properties of the cross-product operator make it useful in math, physics and computer graphics.

Note, if the order of the vectors is switched, the direction of the resulting vector is the opposite.

The formula for the determinant of a two by two matrix.

The magnitude of a vector is given by the square root of the sum of its components squared.

The cross product of two vectors can be calculated using the formal determinant.