1. More on the parametric form

Suppose two different points $P$ and $Q$ are given. A parametric equations for the line through $P$ and $Q$ is found by letting    v$= Q - P$, so that you get the two-point parametric form:

   x$(t) = P + t (Q - P)$.

If you multiply out and condense the terms a different way, you get another way of writing the same thing:

   x$(t) = (1-t) P + t Q$,

which expresses points on the line as linear combinations of the two given points, as $t$ changes. Notice that    x$(0) = P$ and    x$(1) = Q$. The expression can be regarded as a weighted average of the two points, with weights $1-t$ and $t$. If $t = {\frac{1} {2}}$, you have the usual average, the midpoint of the line segment. If $t = {\frac{1} {4}}$ (say), you get the point ${\frac{3} {4}} P + {\frac{1} {4}} Q$, which is a fourth of the way from $P$ to $Q$. In particular, notice that this point is closer to $P$, which has the larger of the two weights.

Observe also that points on the line segment $\overline{PQ}$ are those for which $0 \leq t \leq 1$; if $t < 0$ or $t > 1$ then you get points on the line outside the segment.

The parametric form is useful even for describing individual points on a line segment. If you want the point that is 0.30 of the way from $P$ to $Q$, for example, you can describe it as $P + 0.30(Q-P)$.

Although we're only considering lines in    R$^ 2$ for the moment, the parametric form works in exactly the same way to describe lines in    R$^ 3$. In fact, in    R$^ 3$, the parametric form is by far the best way to describe a line or line segment.