What does the cross product between two vectors represent, and what are some of its properties

See explanation below.

Step-by-step explanation:

Definition

The cross product is a binary operation between two vectors defined as following:

Let two vectors

The cross product is defined as:

The last one is the math definition but we have a geometric interpretation as well.

We define the angle between two vectors a and b and we assume that and we have the following equation:

And then we conclude that the cross product is orthogonal to both of the original vectors.

Some properties

Let a and b vectors

If two vectors a and b are parallel that implies

If then is orthogonal to both a and b.

Let u,v,w vectors and c a scalar we have:

(Distributive property)

Other application of the cross product are related to find the area of a parallelogram for two dimensions where:

And when we want to find the volume of a parallelepiped in 3 dimensions: