Parallelism and Perpendicularity of Two Planes
Two planes are parallel or perpendicular according as the normals to them are parallel or perpendicular.
Hence, the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0
are parallel, if a1 / a2 = b1 / b2 = c1 / c2 and perpendicular, if a1a2 + b1b2 + c1c2 = 0.
Note The equation of plane parallel to a given plane ax + by + cz + d = 0 is given by ax + by + cz + k = 0, where k may be determined from given conditions.
Angle between a Line and a Plane
In Vector Form The angle between a line r = a + λ b and plane r *• n = d, is defined as the complement of the angle between the line and normal to the plane:
sin θ = n * b / | n || b | In Cartesian Form The angle between a line x – x1 / a1 = y – y1 / b1 = z –
z1 / c1 and plane a2x + b2y + c2z + d2 = 0 is sin θ = a1a2 + b1b2 + c1c2 / √a 2 1
+ b 2 1 + c 2 1 √a 2 2 + b 2 2 + c 2 2
Distance of a Point from a Plane Let the plane in the general form be ax + by + cz + d = 0. The distance of the point P( x1, y1, z1) from the plane is equal to
If the plane is given in, normal form lx + my + nz = p. Then, the distance of the point P( x1, y1, z1) from the plane is | lx 1 + my1 + nz1 – p |.