# Find the unit vectors perpendicular to the plane containing the vectors $\overrightarrow{2i}+\overrightarrow{j}+\overrightarrow{k}$ and $\overrightarrow{i}+\overrightarrow{2j}+\overrightarrow{k}$

Toolbox:
• For two vectors $\overrightarrow a \: and \: \overrightarrow b$, the vector product $\overrightarrow a$ x $\overrightarrow b=|\overrightarrow a||\overrightarrow b| \sin \theta \hat n$ with $\hat n \perp$ to $\overrightarrow a \: and \: \overrightarrow b\: and \: \overrightarrow a, \overrightarrow b, \hat n$ forming a right handed system.
• If $\overrightarrow a = a_1\overrightarrow i+a_2\overrightarrow j+a_3\overrightarrow k, \: \overrightarrow b = b_1\overrightarrow i+b_2\overrightarrow j+b_3\overrightarrow k$ then $\overrightarrow a$ x $\overrightarrow b = \begin{vmatrix} \overrightarrow i & \overrightarrow j & \overrightarrow k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
Step 1
Let $\overrightarrow a= 2\overrightarrow i+\overrightarrow j+\overrightarrow k \: and \: \overrightarrow b=\overrightarrow i+2\overrightarrow j+\overrightarrow k$
$\overrightarrow a$ x $\overrightarrow b \: and \: -( \overrightarrow a$ x $\overrightarrow b)$ are vectors $\perp$ to the plane containing $\overrightarrow a\: and \: \overrightarrow b$.
$\therefore$ the unit vectors $\perp$ to the plane containing $\overrightarrow a \: and \: \overrightarrow b$ are
$\pm \hat n = \pm \large\frac{(\overrightarrow a \times \overrightarrow b)}{|\overrightarrow a \times \overrightarrow b|}$
$\overrightarrow a \times \overrightarrow b =\begin{vmatrix} \overrightarrow i & \overrightarrow j & \overrightarrow k \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{vmatrix} = (1-2)\overrightarrow i-(2-1)\overrightarrow j+(4-1) \overrightarrow k$
$= - \overrightarrow i - \overrightarrow j + 3\overrightarrow k$
$\therefore \pm \hat n = \pm \large\frac{(-\overrightarrow i-\overrightarrow j+3\overrightarrow k)}{|-\overrightarrow i-\overrightarrow j+3\overrightarrow k|} =\pm \large\frac{ ( \overrightarrow i+\overrightarrow j-3\overrightarrow k)}{\sqrt{1+1-19}}$
$\pm \large\frac{ \overrightarrow i+\overrightarrow j-\overrightarrow k}{\sqrt{11}}$