Cross Product (Determinant) Formula

This figure illustrates how the cross product operator returns the normal vector to the two input vectors.

Formula

Usage

The cross product formula calculates the cross-product of two vectors (three dimensional). The return value is a vector that is orthogonal (perpendicular) to the input vectors and whose magnitude is equal to the area of the parallelogram between the two input vectors. For example, the cross product of two vectors is given below.

The two input vectors are illustrated in the figure below. Note, since they lie in the plane it makes sense that the output vector would only have a non-zero component.

Cross Product determinant formula usage

To calculate the cross product, 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.

Because the two vectors lie in the coordinate plane and the and components of the resulting vector are zero we know that the output vector is orthogonal to the input vectors. We can also verify the area property by calculating the area of the parallelogram by hand. The area between the two input vectors is shown below.

Cross Product Example

The area can be calculated as area of the rectangle minus the area of the four triangles.

The magnitude of the vector is given below.

Since the area and magnitude are equal this verifies the area property for this example.

Explanation

The formula can be derived by applying the 3 by 3 determinant formula below.

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