Answer
$W^\perp$ is spanned by the vectors $\{(-2,1,0),(-3,0,1)\} $.
Work Step by Step
Let $W$ be the subspace spanned by the vector $(1,2,3)$. Then, $W^\perp$ can calculated as follows
\begin{aligned} W^{\perp} &=\left\{(x, y, z) \in \mathbb{R}^{3} :(x, y, z) \cdot(1,2,3)=0\right\} \\ &=\left\{(x, y, z) \in \mathbb{R}^{3} : x+2 y+3 z=0\right\} \\ &=\left\{(x, y, z) \in \mathbb{R}^{3} : x=-2 y-3 z\right\} \\ &=\{(-2 y-3 z, y, z) : y, z \in \mathbb{R}\} \\ &=\{(-2 y, y, 0)+(-3 z, 0, z) : y, z \in \mathbb{R}\} \\ &=\{y(-2,1,0)+z(-3,0,1) : y, z \in \mathbb{R}\} \\ &=\operatorname{span}\{(-2,1,0),(-3,0,1)\} \end{aligned}
Therefore, $W^\perp$ is spanned by the vectors $\{(-2,1,0),(-3,0,1)\} $.
