The vector space is introduced in Simple tensor algrbra I, several rules are provided in it. To reduce the abstract, we represent the vectors with the scalar coefficients and in , respectively. The composition of two vectors is expanded by,

**Dot product ():**

The *dot product* is also called *scalar product* or *inner product* under Cartesian coordinates and it provides the Euclidean magnitudes of these two vectors and the cosine of the angle between them.

**Cross product ():**

The *cross product* is also called *vector product*, and it represents the area of a parallelogram with the sides of these two vectors. Now, a new composition for two vectors is introduced here.

This is not a tutorial, please refer to the books for tensor algebra.