Question

Question: In exercises 1-6, determine which sets of vectors are orthogonal.

\(\left[ {\begin{align}1\\{ - 2}\\1\end{align}} \right]\), \(\left[ {\begin{align}0\\1\\2\end{align}} \right]\), \(\left[ {\begin{align}{ - 5}\\{ - 2}\\1\end{align}} \right]\)

Short answer

The given set is an orthogonal set.

Step-by-step solution

Definition of an orthogonal set

If \[{{\bf{u}}_i} \cdot {{\bf{u}}_j} = 0\] for \[i \ne j\], then the set of vectors \[\left\{ {{{\bf{u}}_1}, \ldots ,{{\bf{u}}_p}} \right\} \in {\mathbb{R}^n}\] is said to be orthogonal.

Check for orthogonality of vectors

Let the given vectors are, \({u_1} = \left[ {\begin{align}{*{20}{c}}1\\{ - 2}\\1\end{align}} \right]\), \({u_2} = \left[ {\begin{align}{*{20}{c}}0\\1\\2\end{align}} \right]\) and \({u_3} = \left[ {\begin{align}{*{20}{c}}{ - 5}\\{ - 2}\\1\end{align}} \right]\).

First, find :

\(\begin{align}{c}{u_1} \cdot {u_2} = \left( 1 \right)\left( 0 \right) + \left( { - 2} \right)\left( 1 \right) + \left( 1 \right)\left( 2 \right)\\ = 0 - 2 + 2\\ = 0\end{align}\)

Now, find \({u_2} \cdot {u_3}\):

\(\begin{align}{c}{u_2} \cdot {u_3} = \left( 0 \right)\left( { - 5} \right) + \left( 1 \right)\left( { - 2} \right) + \left( 2 \right)\left( 1 \right)\\ = 0 - 2 + 2\\ = 0\end{align}\)

And find \({u_1} \cdot {u_3}\):

\(\begin{align}{c}{u_1} \cdot {u_3} = \left( 1 \right)\left( { - 5} \right) + \left( { - 2} \right)\left( { - 2} \right) + \left( 1 \right)\left( 1 \right)\\ = - 5 + 4 + 1\\ = 0\end{align}\)

Since all the pairs are orthogonal. Hence, the given set is an orthogonal set.