Question

How do you find all unit vectors normal to the plane which contains the points `(0,1,1),(1,−1,0)`, and `(1,0,2)`?

Answer 1

`hat n = {-3/sqrt[14], -sqrt[2/7], 1/sqrt[14]}`

Explanation:

Given three non aligned points there is an unique plane which contains them.

`p_1={0,1,1}`
`p_2={1,-1,0}`
`p_3 ={1,0,2}`

`p_1,p_2,p_2` define two segments

`p_2-p_1` and `p_3-p_1` parallel to the plane which contains `p_1,p_2,p_3`.

The normal to them is also the normal to the plane so

`hat n = ((p_2-p_1) xx (p_3-p_1))/abs( (p_2-p_1) xx (p_3-p_1)) = {-3/sqrt[14], -sqrt[2/7], 1/sqrt[14]}`