Question

How do you find the set of parametric equations for the line through the point P(2,-1,0) which is perpendicular to the plane described by the general equation -x+z=2?

Answer 1

`vec l(s)= <2,-1,0> + s <-1,0,1>`

Explanation:

the plane `pi: - x + z = 2` has normal vector `vec n = <-1,0,1>`

This is because for any plane with normal `vec n`, and on which lies point `vec r_o`, we can say that `(vec r - vec r_o)* vec n = 0`, where `vec r = < x, y, z>`.

Or `vec r * vec n = vec r_o* vec n = const`. And here we have `< x , y, z > * <-1,0,1> = 2`

So the line through P(2,-1,0), and normal to `pi`, parameterised in terms of `s` is:

`vec l(s) = vec l_o + s vec d`

`= <2,-1,0> + s <-1,0,1>`