Orthonormal Basis Vectors

Let there be n vectors in a set of basic vectors where the vectors are n-dimensional.

We say these basis vectors are orthonormal if the following two rules are true:

  1. The length of every vector is 1.
  2. For any vector in the basis set, it forms a 90 degree angle with every other vector in the basis set.

If we are in a 3D vector space and our basis vectors are \hat e_1, \hat e_2, \hat e_3 then the following 9 equations are true:

\begin{matrix} \hat e_1 \cdot \hat e_1 = 1 & \hat e_1 \cdot \hat e_2 = 0 & \hat e_1 \cdot \hat e_3 = 0 \\ & & \\ \hat e_2 \cdot \hat e_1 = 0 & \hat e_2 \cdot \hat e_2 = 1 & \hat e_2 \cdot \hat e_3 = 0 \\ & & \\ \hat e_3 \cdot \hat e_1 = 0 & \hat e_3 \cdot \hat e_2 = 0 & \hat e_3 \cdot \hat e_3 = 1 \end{matrix}