Question

What is the unit vector that is normal to the plane containing <0,2,0> and <-1,1,1>?

Answer 1

`hat n= 1/( sqrt2) ((1),(0),(1))`

Explanation:

the vector cross product is the way to go

`vec a times vec b = |vec a||vec b| sin theta color(red)(\ hat n_{ab})`

and, mechanically, we get that as the determinant of this matrix

`((hat x, hat y, hat z), (0,2,0), (-1,1,1))`

`implies vec n = hat x (2) - hat y (0) + hat z (2) = ((2),(0),(2))`

so, the unit vector `hat n = 1/sqrt(2^2 + 0^2 + 2^2) ((2),(0),(2)) = 1/( sqrt2) ((1),(0),(1))`