Orthogonal vectors | Improve Society

The scalar or dot product of two vectors $a_1\bold{i} + a_2\bold{j} + a_3\bold{k}$ and $b_1\bold{i} + b_2\bold{j} + b_3\bold{k}$ is $a_1b_1 + a_2b_2 + a_3b_3$ and the vectors are perpendicular or orthogonal if $a_1b_1 + a_2b_2 + a_3b_3 = 0$ . From the point of view of matrices we can consider these vectors as column vectors
$\displaystyle A = \left( \begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array} \right),\ B = \left( \begin{array}{c} b_1 \\ b_2 \\ b_3 \end{array} \right)$
from which it follows that $A^TB = a_1b_1 + a_2b_2 + a_3b_3$ .

This leads us to define the scalar product of real column vectors $A$ and B as $A^TB$ and to define $A$ and $B$ to be orthogonal if $A^TB = 0$ .

It is convenient to generalize this to cases where the vectors can have complex components and we adopt the following definition:

Definition 1. Two column vectors $A$ and $B$ are called orthogonal if $\bar{A}^TB = 0$ , and $\bar{A}^TB$ is called the scalar product of $A$ and $B$ .

It should be noted also that if $A$ is a unitary matrix then $\bar{A}^TA = 1$ , which means that the scalar product of $A$ with itself is 1 or equivalently $A$ is a unit vector, i.e. having length 1. Thus a unitary column vector is a unit vector. Because of these remarks we have the following

Definition 2. A set of vectors $X_1,\ X_2,\ \cdots$ for which
$\displaystyle \bar{X}^T_jX_k = \left\{\begin{array}{cc} 0 & j \ne k \\ 1 & j = k \end{array} \right.$
is called a unitary set or system of vectors or, in the case where the vectors are real, an orthonormal set or an orthogonal set of unit vectors.