Due to the orthogonality of the bases, one has the following relations.

Rectangular coordinates in 3-D:

The base vectors of a rectangular coordinate system are given by a set of three mutually orthogonal unit vectors denoted by , , and that are along the x, y, and z coordinate directions, respectively, as shown in the figure.

The system shown is a right-handed system since the thumb of the right hand points in the direction of z if the fingers are such that they represent a rotation around the z -axis from x to y. This system can be changed into a left-handed system by reversing the direction of any one of the coordinate lines and its associated base vector.

In a rectangular coordinate system the components of the vector are the projections of the vector along the x, y, and z directions. For example, in the figure the projections of vector A along the x, y, and z directions are given by A_x, A_y, and A_z, respectively.