Cross Product (Determinant) Formula

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

Formula

Usage

This formula calculates the cross-product of two three dimensional vectors. The cross product produces 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.

Examples

Example 1

This example shows how to calculate the cross product of the two vectors and given below.

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.

    The cross product of the two vectors and is given by the vector above.

Explanation

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

Properties

The cross product produces 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 two vectors and both lie in the plane.

The tow vectors are illustrated in the graphic below.

Cross Product determinant formula usage

From the first propert, the cross product of the two vectors should produce a vector with only a non-zero component. Calcuating the cross product for these vectors demonstrates that this is the case for this example.

The second property of the cross product is that the magnitude of the cross product is equal to the area between the two vectors. The area between the two vectors is illustrated in the graphic below.

Cross Product Example

The area between the two vectors 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.

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