Cross Product Operator

This figure illustrates how the cross product operator returns the normal vector to the two input vectors.
Figure 1: Cross Product Operator

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.

Cross Product Example

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

Cross Product Example

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.

Formulas

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.

Cross Product (Determinant)

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.

Steps

  1. Set up the formula.

  2. Substitute the components of the vectors into the formula.

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

  4. Change to the vertical notation if desired.

Cross Product (Magnitude)

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

Steps

  1. Set up the formula.

  2. Apply the magnitude of vector formula.

Properties

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

Orthogonal Property

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

Area Property

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