What is the unit vector that is normal to the plane containing <0,2,0> and <-1,1,1>?

Aug 2, 2016

$\hat{n} = \frac{1}{\sqrt{2}} \left(\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right)$

Explanation:

the vector cross product is the way to go

$\vec{a} \times \vec{b} = | \vec{a} | | \vec{b} | \sin \theta \textcolor{red}{\setminus {\hat{n}}_{a b}}$

and, mechanically, we get that as the determinant of this matrix

$\left(\begin{matrix}\hat{x} & \hat{y} & \hat{z} \\ 0 & 2 & 0 \\ - 1 & 1 & 1\end{matrix}\right)$

$\implies \vec{n} = \hat{x} \left(2\right) - \hat{y} \left(0\right) + \hat{z} \left(2\right) = \left(\begin{matrix}2 \\ 0 \\ 2\end{matrix}\right)$

so, the unit vector $\hat{n} = \frac{1}{\sqrt{{2}^{2} + {0}^{2} + {2}^{2}}} \left(\begin{matrix}2 \\ 0 \\ 2\end{matrix}\right) = \frac{1}{\sqrt{2}} \left(\begin{matrix}1 \\ 0 \\ 1\end{matrix}\right)$