# What is the unit vector that is orthogonal to the plane containing  <0, 4, 4>  and  <1, 1, 1> ?

Dec 12, 2016

The answer is =〈0,1/sqrt2,-1/sqrt2〉

#### Explanation:

The vector that is perpendicular to 2 other vectors is given by the cross product.

〈0,4,4〉x〈1,1,1〉= | (hati,hatj,hatk), (0,4,4), (1,1,1) |

$= \hat{i} \left(0\right) - \hat{j} \left(- 4\right) + \hat{k} \left(- 4\right)$

=〈0,4,-4〉

Verification by doing the dot products

〈0,4,4〉.〈0,4,-4〉=0+16-16=0

〈1,1,1〉.〈0,4,-4〉=0+4-4=0

The modulus of 〈0,4,-4〉 is =∥〈0,4,-4〉∥

$= \sqrt{0 + 16 + 16} = \sqrt{32} = 4 \sqrt{2}$

The unit vector is obtained by dividing the vector by the modulus

=1/(4sqrt2)〈0,4,-4〉

=〈0,1/sqrt2,-1/sqrt2〉