
A plane through the origin has an equation of the
form Ax + By + Cz = 0, where [A,B,C] is not [0,0,0].
The triple
[uA,uB,uC] will determine the same plane as long as u is not 0.
For example, [1,0,1] determines the plane with
equation X + Z = 0. This plane is also determined by [3,0,3].

A line through the origin has the equation of the
form (X,Y,Z) = (a,b,c) t where (a,b,c) is not (0,0,0).
The
triple (ua,ub,uc)will determine the same line as long as u
is not 0.
For example, (1,0,1) determines the line with
equation (X,Y,Z)=(1,0,1)t. This point is also determined by (3,0,3).

A Ppoint has the equation (X,Y,Z) = (a,b,c) t
where (a,b,c) is not (0,0,0).The triple (ua,ub,uc) will determine
the same line as long as u is not 0.
We'll call <a,b,c> homogeneous coordinates
of the P point.
For example, <1,0,1> are homogeneous coordinates
for the Ppoint determined by the line with equation (X,Y,Z) = (1,0,1)
t. <3,0,3> are homogeneous coordinates for the same Ppoint.
 A Pline has an equation of the form Ax + By + Cz
= 0, where [A,B,C] is not [0,0,0].
The triple [uA,uB,uC] will determine the same plane as
long as u is not 0.We'll call [A,B,C] homogeneous coordinates
of the Pline.
For example, [1,0,1] are homogeneous coordinates
for the Pline determined by the plane with equation X + Z = 0.
[3,0,3] are homogeneous coordinates for the same Pline.

A Ppoint lies on a P line or a P line passes through
the the Ppoint if and only if Aa+Bb+Cc= 0 where [A,B,C] are homogeneous
coordinates for the Pline and <a,b,c> are homogeneous coordinates for
the Ppoint.
For example, the Ppoint <1,0,1> lies on the
Pline [1,0,1].