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Chapter 10. Vector Functions

This page contains links to all resources mentioned in Chapter 10 of the textbook. It begins with several ways to graph vector functions for free on the web, and concludes with how to graph them within POV-Ray scenes.

I recommend them in the order shown; GeoGebra is the best. The two that follow it require Java, but do create interactive plots. The final one does not require Java, but creates only a static image.

GeoGebra

GeoGebra (named for "geometry" and "algebra" together) is a very powerful tool and is free to use online. Follow the link above to its 3D graphing tool, and just type in a function, such as f(t)=(3t,2t+1,sin(6t)) to graph it.

GeoGebra automatically converts this to its own internal code for you, which looks like the following, and lets you specify the lower and upper bounds on the variable t.

f = Curve((3t,2t+1,sin(6t)),t,-10,10)

You can type such code in directly, with or without the f =, if you prefer.

You can also use GeoGebra to graph 2D parametric plots.

CalcPlot3D

A useful tool for graphing vector functions and other kinds of 3D objects. Although this applet was created for use in calculus classes, it is useful to us as well.  Use the following procedure to graph a vector function in CalcPlot3D.

  1. Erase the default shape that appears, by unchecking the box next to Function 1 and clicking the Graph button immediately above it.
  2. Add a parametric curve by clicking the Graph menu and choosing Add a Space Curve.
  3. In the three blanks provided, enter the x, y, and z components of the vector function, using t as the parameter. The default bounds for t (from -10 to 10) may be sensible for your function, but you can change them.
  4. Click Graph (on the popup window into which you typed the parametric equations).
  5. Click and drag to view from different angles.

Sage

A powerful mathematics tool that you can use on your own computer or on the web.  Here is a link to a webpage that evaluates Sage code and shows you the result immediately.  Type in code like the following to graph a vector function.  (Replace the three components of the vector function with any three vector function components.)  The 0 and 2\pi are the bounds on t.

var('t')
parametric_plot3d([cos(t),sin(t),t],(0,2*pi))

To see that example plotted, click here.

Wolfram|Alpha

Like Sage, this online math tool does far more than just graphing vector functions, but we can use it for that, too.  In the input box, type input like the following.

parametric plot (t*cos t, t*sin t,t), t=0..6pi

To see that example plotted, click here.

Graphing vector functions in POV

Download this file by clicking vector-functions.inc. Instructions for using it appear in the textbook, and you can download vector-demo.pov to see how I have used it in a demo.

Review of Derivatives

If it's been awhile since you've had differential calculus (what many colleges call "Calculus I") then here are some online review resources available to you.  There are a great many, and this is just a selection.