Question

How do you find a unit vector that is perpendicular to both `-2i + j + 3k` and `i - j - 2k`?

Answer 1

`hat e = {1/sqrt(3) hat i,-1/sqrt(3) hat j,1/sqrt(3) hat k}`

Explanation:

Given two vectors `vec u` and `vec v`, if they are linearly indepedend we can a unit vector perpendicular to both as follows.

`hat e = (vec u xx vec v)/norm(vec u xx vec v) = -(vec v xx vec u)/norm(vec u xx vec v)`

In the present case

`{-2hat i, 1 hat j, 3 hat k}xx{1 hat i, -1 hat j, -2 hat k} = {1 hat i,-1 hat j,1 hat k}`

then

`hat e = {1/sqrt(3) hat i,-1/sqrt(3) hat j,1/sqrt(3) hat k}`