Cross product

The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that

The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis.

Figure 1. The area of a parallelogram as the magnitude of a cross product

Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value:

Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive).

More generally, the cross product obeys the following identity under matrix transformations:

The product rule of differential calculus applies to any bilinear operation, and therefore also to the cross product:

The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as

The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula

Other identities relate the cross product to the scalar triple product:

The same result is found directly using the components of the cross product found from:

This notation is also often much easier to work with, for example, in epipolar geometry.

From the general properties of the cross product follows immediately that

The above-mentioned triple product expansion (bac–cab rule) can be easily proven using this notation.

The word "xyzzy" can be used to remember the definition of the cross product.

Similarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may be helpful for remembering the correct cross product formula.

The cross product has applications in various contexts. For example, it is used in computational geometry, physics and engineering. A non-exhaustive list of examples follows.

The cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space.

The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.

The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints.

Because the cross product may also be a polar vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a vector triple product involving three polar vectors is a polar vector.

There are several ways to generalize the cross product to higher dimensions.

The exterior product and dot product can be combined (through summation) to form the geometric product in geometric algebra.

These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity.