Cross Product

The cross product between two Vectors returns another vector. A vector where has two properties.

1. is perpendicular to both and . In terms of the dot product, we can say that

This also means that the cross product only works in three dimensions. In four dimensions, there are infinitely many vectors perpendicular to two vectors (Think of it as a line of perpendiculars). 2. The length of is a measure of how far apart and are pointing, augmented by their magnitudes.

It looks like the dot product with a sine instead of a cosine. Because of this, when the angle is 90 degrees, the cross product is largest. This makes the dot product and cross product complement each other.

One interpretation of the length of is very useful.

The base of the parallelogram has length and the height has length . The area of the parallelogram is the magnitude of the cross product (base times height).