maybe i'm missing something, here....why doesn't (4,0,1) + t(2,-2,0) work?
I have a normal vector (2,-2,0) of the a plane and a point on a line orthogonal to the plane, (4,0,1). So my goal is to try and find the parametric description of this line.
now to do that I need a vector v, but how can I find this vector with the given info?
the parametric representation is given by p + tv where p is a point and v is a vector. I need another point on this line to find the vector. Since this line is orthogonal to the plane I know that this vector should be parallel to the normal vector of the plane. But im having difficulty trying to find another point on this line.
the normal vector usually doesn't lie on the line going through a given point, it's just parallel to the line (there's a whole plane's worth of such lines, like a bundle of straws all lined up together).
but if we specify a point (any point) then we know which parallel line. "orthogonal to" and "normal" have almost the same meaning (usually, a normal vector is associated with some particular point in the plane). thus, every orthogonal vector to a plane (it's line direction) is parallel to a normal vector to a plane.