Find set of vectors orthogonal to $\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}$ $\mathbf{Question:}$ Find set of vectors orthogonal to $\begin{bmatrix}
1 \\
1 \\
1 \\
\end{bmatrix}$
$\mathbf{My\ attempt:}$
The vector is in $R^3$ so we can let vector $\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix} $ represent orthogonal vectors.
$$\begin{bmatrix}
1 \\
1 \\
1 \\
\end{bmatrix}
.
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix} 
= 
\begin{bmatrix}
0 \\
0 \\
0 \\
\end{bmatrix}
$$
$$
x_1 + x_2 + x_3 = 0
$$
$$
x_1 = -x_2 - x_3
$$
$$
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix}
=
\begin{bmatrix}
-x_2 - x_3 \\
x_2 \\
x_3 \\
\end{bmatrix}
=
\begin{bmatrix}
-1 \\
1 \\
0 \\
\end{bmatrix}x_2
+
\begin{bmatrix}
-1 \\
0 \\
1 \\
\end{bmatrix}x_3
$$
Therefore, the vectors in subspace $W = span(\begin{bmatrix}
-1 \\
1 \\
0 \\
\end{bmatrix} , \begin{bmatrix}
-1 \\
0 \\
1 \\
\end{bmatrix} )$ are orthogonal to vector $\begin{bmatrix}
1 \\
1 \\
1 \\
\end{bmatrix}$.

The space of vectors orthogonal to $\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}$ can be defined by $span(\begin{bmatrix}
-1 \\
1 \\
0 \\
\end{bmatrix} , \begin{bmatrix}
-1 \\
0 \\
1 \\
\end{bmatrix} )$ where the dimension of the space is 2.
