# How do you find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (4, 0, 2), Q = (7, 3, 5), and R = (7, 3, 8)?

Sep 30, 2016

$\vec{n} = \left(\frac{1}{\sqrt{2}} , - \frac{1}{\sqrt{2}} , 0\right)$

#### Explanation:

The plane normal is proportional to

$\vec{v} = \left(P - Q\right) \times \left(R - Q\right) = \left\{- 9 , 9 , 0\right\}$ so

$\vec{n} = \lambda \vec{v}$ now choosing $\lambda = - \frac{1}{\sqrt{{9}^{2} + {9}^{2} + {0}^{2}}}$ we have

$\vec{n} = \left(\frac{1}{\sqrt{2}} , - \frac{1}{\sqrt{2}} , 0\right)$