Recall from the Section 2 that the
coordinates of the line passing through two points
and
is given by

Notice that these three coordinates are just the determinants of the three submatrices of the following matrix:

taken in the appropriate order and given the appropriate sign.

The procedure is similar in
.
The coordinates of the
line
passing through two points
and
is given by
the determinants of the six
submatrices of the following matrix:

In other words, , where

These coordinates

where ,

which can be derived by noting that the determinant is identically zero.

Where does the magic number six come from? That is, why do we need
six parameters to represent a line in
? Interestingly, it turns
out that it takes
(^{n+1}_{k}) parameters
to represent an
entity defined by *k* points in a space requiring *n*+1 parameters for
each point (To see this, count the number of submatrices in the
matrix above).
For example, in
a point requires
(^{3}_{1}) = 3 parameters, and a line (which is defined by
two points) also requires
(^{3}_{2}) = 3 parameters. In
,
a point requires
(^{4}_{1}) = 4 parameters, a line
(^{4}_{2}) = 6 parameters, and a plane
(^{4}_{3}) = 4 parameters.