1.6.1—Plane equations

A plane through point a with normal vector n has equation n·(r−a)=0, or equivalently n·r=d. The normal is perpendicular to every direction lying in the plane.

To find a normal, use a cross product of two non-parallel direction vectors in the plane. Check a candidate point by substitution and keep vector notation consistent.

A plane through (1,0,2) with normal (2,−1,3) satisfies 2(x−1)−y+3(z−2)=0.

A vector lying in the plane is perpendicular to the normal, not parallel to it; one point alone does not determine a unique plane.