# Find the orthogonal complement of W and give a basis

Find the orthogonal complement of W and give a basis for
$W=\{\begin{bmatrix}x\\ y\\ z\end{bmatrix}:x=\frac{1}{2}t,\ y=-\frac{1}{2},\ z=2t\}$

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Walker Funk
So we need to find two linearly independent vectors orthogonal to
$\begin{bmatrix}\frac{1}{2}\\-\frac{1}{2}\\2\end{bmatrix}$
$\begin{bmatrix}1\\-1\\4\end{bmatrix}$ (or to avoid fractions). To do so, we can just use the definition of orthogonality:
$$\displaystyle{\left({x},{y},{z}\right)}\cdot{\left({1},-{1},{4}\right)}={x}-{y}+{4}{z}={0}$$
Let's let x=1 and y=0. Then we see that $$\displaystyle{z}=-{\frac{{{1}}}{{{4}}}}$$. So
$\begin{bmatrix}1\\0\\-\frac{1}{4}\end{bmatrix}$
is in W. Now let x=0 and y=1. Then $$\displaystyle{z}={\frac{{{1}}}{{{4}}}}$$. So
$\begin{bmatrix}0\\1\\\frac{1}{4}\end{bmatrix}$
is also in W. Thus
$W=\{\begin{bmatrix}1\\0\\-\frac{1}{4}\end{bmatrix},\begin{bmatrix}0\\1\\\frac{1}{4}\end{bmatrix}\}$
where
$\{\begin{bmatrix}1\\0\\-\frac{1}{4}\end{bmatrix},\begin{bmatrix}0\\1\\\frac{1}{4}\end{bmatrix}\}$
Is a basis for W