**Theorem
2.2**: (The parametric form of the Ruler Axiom) Let* t* be a real
number. Let *A* and *B* be two points. Let

and

Let

using vector addition and scalar multiplication of points. Then

and

Which is to say that, if *C* is a point on the line segment between *A* and *B*, that

|*AB*| = |*AC*| + |*CB*|

**Proof**: Let's first compute, using the
distance formula

=|*t*||*AB*|

Similarly,

= (1 - *t*)|*AB*|

Thus, since *C* is between *A* and *B*, 0 < *t* < 1, |*t*| = *t*, and |1 - *t*| = 1 - *t*, so

|*AC*| + |*CB*|

= *t*|*AB*| + (1 - *t*)|*AB*|

= |*AB*|