# What is the unit vector that is orthogonal to the plane containing  (32i-38j-12k)  and  (41j+31k) ?

Jul 11, 2016

$\hat{n} = \frac{1}{\sqrt{794001}} \left[- 343 \vec{i} - 496 \vec{j} + 656 \vec{k}\right]$

#### Explanation:

The cross product of two vectors produces a vector orthogonal to the two original vectors. This will be normal to the plane.

$| \left(\vec{i} , \vec{j} , \vec{k}\right) , \left(32 , - 38 , - 12\right) , \left(0 , 41 , 31\right) | = \vec{i} | \left(- 38 , - 12\right) , \left(41 , 31\right) | - \vec{j} | \left(32 , - 12\right) , \left(0 , 31\right) | + \vec{k} | \left(32 , - 38\right) , \left(0 , 41\right) |$

$\vec{n} = \vec{i} \left[- 38 \cdot 31 - \left(- 12\right) \cdot 41\right] - \vec{j} \left[32 \cdot 31 - 0\right] + \vec{k} \left[32 \cdot 41 - 0\right]$

$\vec{n} = - 686 \vec{i} - 992 \vec{j} + 1312 \vec{k}$

$| \vec{n} | = \sqrt{{\left(- 686\right)}^{2} + {\left(- 992\right)}^{2} + {1312}^{2}} = 2 \sqrt{794001}$

$\hat{n} = \frac{\vec{n}}{| \vec{n} |}$

$\hat{n} = \frac{1}{\sqrt{794001}} \left[- 343 \vec{i} - 496 \vec{j} + 656 \vec{k}\right]$