What is the easiest way to decided if two planes are coincident?

Question

I was given two planes in the form ax + by + cz = d
If you have their normals (a,b,c),

Say, u = (2,-1,2)
and v = (1,2,-3)

Can you easily tell if these are the same plane?

________________

u.v = -6

and

u is not a non 0 multiple of v so therefore not parallel.

What is the last test to see if the planes are coincidental?

Answer 1

(The planes are coincident or parallel) IF AND ONLY IF (the normals are parallel*)

*Or coincident.

One way to distinguish between the coincident and parallel case is working out the equations.

If one has equation Ax + By + Cz = D and the other has equation Ax + By + Cz = E, they are coincident if D=E and parallel otherwise.

Another way is to take a point on one plane and a point on the other, view them as vectors, and subtract one from the other. If that's perpendicular to a normal, then they lie in the same plane and the planes coincide. Otherwise, they don't.

Answer 2

If the normals are not parallel the planes are not coincidental.

Two planes with equations in the form

ax + by + cz = d, Ax + By + Cz = D

are coincidiental if (and only if) one equation is a multiple of the other, i.e. there is a constant k such that:

ak = A, bk = B, ck = C, and dk = D

Answer 3

The planes have to be one of coincident, parallel, or distinct. That is all there is. If the normal vectors of the planes are not parallel, then the planes are distinct. The cross product of parallel vectors is zero. If the cross product of the normal vectors is not zero, then the planes are distinct. n1 = <1, 2, -4> n2 = <2, -2, -5> n1 X n2 = <1, 2, -4> X <2, -2, -5> = <-18, -3, -6> The cross product is not the zero vector. Therefore the planes are distinct.

Answer 4

if the two planes are flying towards each other in opposite directions, they may be coincidental, but ground controllers better change the altitude of one or both or they won't be coincident very long.