## 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 **v****r** = d where **v** is the vector (a, b, c) and **r** is (x, y, z).

Thus solutions to it all have the same value of their component in the direction
of **v**, and are not determined in directions **perpendicular to, normal to**, or **orthogonal to v **(all these words mean the same
thing). Thus **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.