# Vector Perpendicular to Plane

1. May 22, 2012

### artkingjw

the way to find out if a vector is perpendicular to a plane

the usual way is to find the normal to the plane, then find the cross product of the normal with the vector you are given. if it is 0 then perpendicular, else it is not correct?

what i want to know is, can you also, put the the plane and vector into a matrix, and solve for the constants? if no solution exists, the vector does not intersect the plane thus is parallel? i have a feeling i'm wrong...

2. May 22, 2012

### chiro

Hey artkingjw and welcome to the forums.

If your plane equation is in the form n . (r - r0) = 0, then your vector is perpendicular to the plane if Unit(n) . Unit(d) = 1 or -1 where Unit(x) = x/||x|| where ||x|| is the length of the vector.

If your plane is written in the form ax + by + cz + d = 0, then form a vector n = (a,b,c) take x = n/||n|| and then calculate Unit(d) . x and test if its -1 or +1.

Also . means the dot product or inner product for cartesian space which is simply x1y1 + x2y2 + x3z3 in 3D space for (x1,x2,y3) and (y1,y2,y3) vectors.