Two parametric lines

An interactive 3D plot

See questions below.

This worksheet is intended to help you explore parametric lines in three-space.

The worksheet allows you to specify two points, \(P\) and \(Q\), two vectors v and w, and two parameters \(t\) and \(s\) (by default, \(t = 1\) and \(s = 1\)). Initially, four vectors will be plotted:

  • v (in red) based at point \(P\),
  • w (in green) based at point \(Q\),
  • \(t\)v (in blue) based at point \(P\),
  • \(s\)w (in blue) based at point \(Q\)

You should observe the effects of varying \(t\) and \(s\). If \(t\) is allowed to take on any real number value, the terminal points of \(\overrightarrow{p} + t\overrightarrow{v}\) describe a line. Note that you have the option of plotting the position vector \(\overrightarrow{p}\) corresponding to point \(P\) and the vector \(\overrightarrow{p} +t\overrightarrow{v}\). You also have the option of plotting the "entire line", but you'll need to zoom in to see the details if you do so.

So without further ado, click the “Evaluate” button below (the actual Sage code is included in the box only in case you're interested; you don't need to modify the code).

Questions

  • Can you create two lines that intersect?
  • Can you make the two lines exactly equal? (Without simply setting \(P = Q, v = w\))

Work on understanding the following components:

  • the point \(P\)
  • the vector \(\overrightarrow{p}\)
  • the vector \(\overrightarrow{p} + t\overrightarrow{v}\) for a specific value of \(t\)
  • the terminal point of the vector \(\overrightarrow{p} + t\overrightarrow{v}\) for a specific value of \(t\)
  • the terminal points of the vectors \(\overrightarrow{p} + t\overrightarrow{v}\) for all values of \(t\)
  • Can you make the two lines parallel? If so, how?
  • Can you make the two lines perpendicular but not intersecting? If so, how?