5.2 Representations of a Line or Plane: Preliminary Remarks

A single linear equation can be used to solve for one variable in terms of the others. It reduces the dimension of the set of its solutions by 1.

Thus
the points in a plane in 3 dimensions will be the solutions of
one linear equation. In two dimensions one linear equation determines a line,
while
in three dimensions two equations are needed to determine a line.

A linear equation of the form
$ax+by+cz=d$
can be written as the dot product
$\stackrel{\u27f6}{v}\xb7\stackrel{\u27f6}{r}=d$
where
$\stackrel{\u27f6}{v}$
is the vector
$(a,b,c)$
and
$\stackrel{\u27f6}{r}$
is
$(x,y,z)$
.

Thus solutions to it all have the same value of their component in the direction of
$\stackrel{\u27f6}{v}$
, and are not determined in directions
perpendicular to, normal to
, or
orthogonal to
$\stackrel{\u27f6}{v}$
(all these words mean the same thing). Thus
$\stackrel{\u27f6}{v}$
is normal to a vector pointing from one solution to another.

A line can be characterized by giving the coordinates of two points on it
(in any space) and a plane
by giving the coordinates of three points that
don't all lie on a single line.

Further we can describe
a line by giving any point on it, and a vector
that points in its direction.

Likewise we can describe the
points on a plane by giving a point and two
linearly independent vectors starting from that point that lead to other points
in the plane.

We now examine the relations between the various characterizations of lines and planes.