α
is a plane in the space and we've been told in school that its equation in canonical form is:
ax+by+cz+d=0
But this doesn't tell us nothing about the plane characteristics and no-one explained us how this equation is formed.
Plane formation
A plane in the space could be identified by a point and 2 non-parallel vectors.
P=(x0,y0,z0)∈αu,v∈α
u,v
have magnitude, direction, and orientation. It's possible to write:
u=PQv=PR
Every other point
X
that belongs to
α
could be written as a linear combination of these elements starting from the origin:
OX=OP+s∗u+t∗vX=(x,y,z)∈αs∈Rt∈R
This clarifies why it is possible to say:
A plane in the space could be identified by a point and 2 non-parallel vectors.
Parametric equations
X
can be described by its parametric equations:
x=x0+m1∗s+m2∗ty=y0+n1∗s+n2∗tz=z0+p1∗s+p2∗tu=(m1,n1,p1)v=(m2,n2,p2)
The goal is to solve this system over
s
and
t
reducing it to a linear equation that is satisfied for the points belongings to the plane.
A bit of math
t=m2x−x0−m1∗st=n2y−y0−n1∗st=p2z−z0−p1∗s
Equal over
t
:
m2x−x0−m1∗s=n2y−y0−n1∗sm2x−x0−m1∗s=p2z−z0−p1∗s
t
is gone, let's target
s
m2x−x0−m2m1∗s=n2y−y0−n2n1∗sm2x−x0−m2m1∗s=p2z−z0−p2p1∗s
m2x−x0−n2y−y0=(m2m1−n2n1)∗sm2x−x0−p2z−z0=(m2m1−p2p1)∗s
A=m2x−x0B=n2y−y0C=m2m1D=n2n1E=p2z−z0F=p2p1
A−B=s∗(C−D)A−E=s∗(C−F)
s=C−DA−Bs=C−FA−E
(A−B)(C−F)=(A−E)(C−D)AC−AF−BC+BF=AC−AD−CE+DE−AF−BC+BF=−AD−CD+DEAF−AD+BC−BF−CE+DE=0A∗(F−D)+B∗(C−F)+E∗(D−C)=0
m2x−x0(p2p1−n2n1)+n2y−y0(m2m1−p2p1+p2z−z0∗(n2n1−m2m1)=0
a=m2p2p1−n2n1b=n2m2m1−p2p1c=p2n2n1−m2m1
a∗(x−x0)+b∗(y−y0)+c∗(z−z0)=0ax+by+cz−ax0−by0−cz0=0d=−ax0−by0−cz0ax+by+cz+d=0
w=(a,b,c)
This is the directional vector of the plane, orthogonal to it.
Geometry
To prove that
w
is orthogonal to the plane, it should be orthogonal to both
u
and
v
.
The scalar product should be 0:
w⋅u=0w⋅v=0
w⋅u=a∗m1+b∗n1+c∗p1=m2p2p1−n2n1∗m1+n2m2m1−p2p1∗n1+p2n2n1−m2m1∗p1
w⋅u=p2p1m2m1−n2n1m2m1+m2m1n2n1−p2p1n2n1+n2n1p2p1−m2m1p2p1=0
w⋅v=a∗m2+b∗n2+c∗p2=m2p2p1−n2n1∗m2+n2m2m1−p2p1∗n2+p2n2n1−m2m1∗p2=0
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