. . Distance of a point from a plane (a) Equation of a plane in vector form Theorem . The perpendicular distance from a point with position vector u to the plane r n is given by u n δ Proof Let A be the point whose position vector is u .
Vector - - Applications of Vector Algebra Let F be the foot of the perpendicular from the point A to the plane r n . The line joining F and A is parallel to the normal vector n hence its equation is r u tn . But F is the point of intersection of the line r u tn and the given plane r n . If r is the position vector of F , then u t n for some t ∈ , and r n .Eliminating r we get u t n which implies u n Now, FA u u t n = u n t n = Therefore, the length of the perpendicular from the point A to the given plane is δ = u n u n FA = The position vector of the foot F of the perpendicular AF is given by r = u t n or r = u n u + (b) Equation of a plane in Cartesian form In Caretesian form if ( A x y z is the given point with position vector u and ax by cz is the Cartesian equation of the given plane, then ˆ u x i y j z k ai bj ck .
Therefore, using these vectors in u n δ , we get the perpendicular distance from a point to the plane in Cartesian form as ax by cz ax by cz δ Remark The perpendicular distance from the origin to the plane ax by cz = is given by δ = Example . Find the distance of a point ( , , ) from the plane