What is the unit vector that is orthogonal to the plane containing  (3i + 2j - 3k)  and  ( i - j + k) ?

Feb 29, 2016

$\setminus {\hat{n}}_{A B} = - \frac{1}{\setminus} \sqrt{62} \left(\setminus \hat{i} + 6 \setminus \hat{j} + 5 \setminus \hat{k}\right)$

Explanation:

The unit vector perpendicular to the plane containing two vectors $\setminus \vec{{A}_{}}$ and $\setminus \vec{{B}_{}}$ is :

$\setminus {\hat{n}}_{A B} = \setminus \frac{\setminus \vec{A} \setminus \times \setminus \vec{B}}{| \setminus \vec{A} \setminus \times \setminus \vec{B} |}$

\vec{A_{}} = 3\hat{i}+2\hat{j}-3\hat{k}; \qquad \vec{B_{}} = \hat{i}-\hat{j}+\hat{k};

\vec{A_{}}\times\vec{B_{}} = -(\hat{i}+6\hat{j}+5\hat{k});
$| \setminus \vec{{A}_{}} \setminus \times \setminus \vec{{B}_{}} | = \setminus \sqrt{{\left(- 1\right)}^{2} + {\left(- 6\right)}^{2} + {\left(- 5\right)}^{2}} = \setminus \sqrt{62}$

$\setminus {\hat{n}}_{A B} = - \frac{1}{\setminus} \sqrt{62} \left(\setminus \hat{i} + 6 \setminus \hat{j} + 5 \setminus \hat{k}\right)$.