Two Orthogonal Vectors
Definition: Two vectors are orthogonal if they are perpendicular to each other. That is, the dot product of the two vectors is zero.
The following is an example of two orthonormal vectors.
\( \vec{V_1} =\left[\begin{array}{ccc}
1\\
0\\
-1\end{array}
\right]\), \( \vec{V_2} =\left[
\begin{array}{ccc}
1\\
3\\
1
\end{array}
\right] \)
That is, \( (1 \times 1) + (0 \times 3) +(-1 \times 1) = 1+0-1=0\).
A Set of Mutually Orthogonal Vectors
Definition: A set of vectors { \( \vec{v_1}, \vec{v_2}, …, \vec{v_n}, \) } are mutually orthogonal if every pair of vectors is orthogonal.
The following is an example of a set of mutually orthogonal vectors.
\( \vec{V_1} =\left[\begin{array}{ccc}
1\\
0\\
-1\end{array}
\right]\), \( \vec{V_2} =\left[
\begin{array}{ccc}
1\\
3\\
1
\end{array}
\right] \), \( \vec{V_3} =\left[
\begin{array}{ccc}
1\\
-\frac{2}{3}\\
1
\end{array}
\right]\)
That is,
for \( \vec{v_1} \) and \( \vec{v_2}\), \( (1 \times 1) + (0 \times 3) +(-1 \times 1) = 1+0-1=0\).
for \( \vec{v_1} \) and \( \vec{v_3}\), \( (1 \times 1) + (0 \times -\frac{2}{3}) +(-1 \times 1) = 1+0-1=0\).
for \( \vec{v_2} \) and \( \vec{v_3}\), \( (1 \times 1) + (3 \times -\frac{2}{3}) +(1 \times 1) = 1-2+1=0\).
Thus, we can say that the set of { \( \vec{v_1}, \vec{v_2}, \vec{v_3}, \) } is a set of mutually orthogonal vectors.
An Orthogonal Set
Definition: A set of vectors is orthonormal if every vector in the set has magnitude 1 and the vectors in the set are mutually orthogonal.
As discussed, the following vectors in the set are mutually orthogonal, but their magnitudes are not equal to 1. We can standardize them to make their magnitudes to be all 1.
\( \vec{U_1} = \frac{\vec{V_1}}{|\vec{V_1}|} =\frac{1}{\sqrt{2}}\left[
\begin{array}{ccc}
1\\
0\\
-1
\end{array}
\right] \)
\( \vec{U_2} = \frac{\vec{V_2}}{|\vec{V_2}|} =\frac{1}{\sqrt{11}}\left[
\begin{array}{ccc}
1\\
3\\
1
\end{array}
\right] \)
\( \vec{U_3} = \frac{\vec{V_3}}{|\vec{V_3}|} =\frac{3}{\sqrt{22}}\left[
\begin{array}{ccc}
1\\
-\frac{2}{3}\\
1
\end{array}
\right] \)
Proposition: In an orthogonal set of non-zero vectors, the vectors are linearly independent.
Two or more vectors are said to be linearly independent if none of the vectors can be written as a linear combination of the other vectors.
In contrast, if at least one of the vectors is a linear combination of the other vectors, they are said to be linearly dependent.