Given 
with 
,
we define the line on a and b,
denoted by
,
to be the set of all
unit-sum linear combinations of a and b, i.e.
We say that is a line through the origin if 
.
It is possible to solve
for 
obtaining
.
Substituting this in the expression
we obtain a parametric form
for the points of the line on a and b :
In the above notation, points a and b are
understood as fixed data relative to the discussion.
If it becomes necessary to specify the points
as data parameters,
we can write 
instead of 
,
a notational device often seen in statistics literature.
Thus an alternative espression for the line on points
a and b is
Some important subsets of the line on a and b are the half-lines directed from a through b defined by
and the segments between a and b defined by
The segment
is called
.
In terms of the original formulation of ,
the half-line
is the set of unit-sum linear combinations of a and b
with nonnegative combining coefficients for b,
the open segment
is the set of all positive unit-sum linear combinations of a and b,
and
the closed segment
is the set of all convex linear combinations of a and b .
Given vectors 
,
we define the rays determined by
ray origin p and ray direction d
to be
We may rewrite as 
to obtain the equivalent
representations
It is now easy to see that 
,
where the ray direction b-a has been manufactured from the two points
a and b.
Also, 
Similarly, we can rewrite 
as
so that
, where now a second point has been
manufactured by adding the direction d to the point p.
Proof Left as an exercise.
