Parametrization of a line
A line is determined by two points P and Q. The following interactive Concept-Visualization Tool (CVT) illustrates this simple idea (the main purpose of this first CVT is to get you used to manipulating them). As mentioned in the CVT intro, you can drag with the mouse either the red or the green points around to change the line. Dragging elsewhere rotates the picture.
We want to write an equation for the line. However, since we are in three dimensions, we can’t write a single equation like y = 2x - 3 to describe the line. (If you don’t believe me, check out this page.) Instead, we’ll write a parametrization for the line.
Before we continue, we need to change our perspective about the line. First, rather than thinking of a line being determined by two points, we’ll think of it as being determined by a point P and a vector v parallel to the line. (The vector v could simply be the vector from the point P to the point Q.)
Second, we’ll think of the point P as the endpoint of a vector a with tail at the origin. We could visualize the new perspective with the following CVT. It’s essentially identical to the above CVT. You can drag the tips of the arrows to change the line and rotate the whole figure by dragging in other places.
Unfortunately, I haven’t figured out a good way to draw vectors with arrowheads so that the arrowheads are always visible. So, I’ll tend to draw vectors with dots for arrowheads because that works more consistently, as follows.
Let x (pictured in blue below) be a vector that represents another point on the line. (We will refer to the point x as being the endpoint of the vector x when its tail is fixed at the origin.) The point x is on the line if the vector from a to x (i.e, x - a, pictured in cyan below) is parallel to v (in green). In the CVT below, you can change x be dragging it, but it is constrained to be on the line. (Since x is on the line, the vector v is always parallel to x - a, although sometimes both vectors don’t show up as the CVT doesn’t show overlapping vectors well). Recall that you can press Home to restore the CVT to its original perspective.
What does it mean for two vectors to be parallel? (Note that we use the term
“parallel” to include what you might think of as anti-parallel, meaning
pointing in opposite directions.) Two vectors are parallel if one vector
can be expressed as a scalar multiple of the other. So, x - a is parallel
to v if and only if x - a = tv for some t 
R. We usually write this as
x = tv + a.
The idea of the parametrization is that as the parameter t sweeps through all real numbers, x sweeps out the line. In the last CVT on this page, we’ve changed the perspective so that now you control t by moving the point on the slider. To make visualization a little easier, we show only the endpoints of the vectors a and x. Note that you can no longer move x directly (attempting to drag the blue point will just rotate the whole figure). You control the position of x through changing t with the slider.
Notice that when t = 0, then x = a. And when t = 1, x is on top of the endpoint of vector v, which is at position a + v.
If you want something more concrete, you can see examples of finding the parametrization of a line.
