Linear Algebra, Perpendicular to a Plane

Perpendicular to a Plane

If v is the vector [3,5,7] in 3 space, the points in the plane 3x+5y+7z = 0 are precisely the points that are perpendicular to v.  These are all the vectors w such that v.w = 0.

Given any vector, turn its conponents into coefficients, and build the plane perpendicular to that vector.  The points perpendicular to two vectors are the intersection of the two perpendicular planes, and so on.

Conversely, given a plane, turn the coefficients into the components of a vector, and you have the vector that is perpendicular to that plane.

To find the distance from a point q to a plane, let u be a unit vector perpendicular to the plane, consider the line q+tu, find the intersection of the line and the plane (solve for t), and compute the distance between q and this point of intersection.