SUNY Geneseo Department of Mathematics
Monday, February 20
Math 223
Spring 2023
Prof. Doug Baldwin
Problem set 4, question 1C (are Spot the Space Dog and Spacey’s baseball ever at the same place at the same time?)
The key to this is figuring out whether there is a t value that makes the equations for the ball’s path and Spot’s equal. The units in this problem are set up so that using the same t parameter in both lines makes sense. This is different from earlier in-class “do two lines intersect” examples, where we used different parameters for each line.
Problem set 4, question 4 (where do line and plane intersect?)
Use a parametric equation for the line, then plug its x, y, and z functions into the equation of the plane and solve for t.
Based on “Definition of a Vector-Valued Function” and “Graphing Vector-Valued Functions” in section 2.1
A vector valued function is like a set of parametric equations for a curve, in that you plug in values of a parameter (often named t) into functions that give x, y, and z components. Technically these are the components of a vector, but we will generally blur the distinction between vectors and the points at their ends when they start at the origin.
There are 2 and 3 (and more) dimensional vector-valued functions, and graphs of the curves they define.
Curves defined by vector-valued functions have an “orientation,” i.e., a direction the curve advances in as the parameter to the function increases.
I recommend using Mathematica or similar technology to plot vector-valued functions.
Domain restrictions? Domain restrictions are restrictions on the values of the parameter, usually associated with values for which one or more component functions aren’t defined. For example r(t) = ⟨ √t, 1/t ⟩ , which is only defined when t > 0. So t > 0 is a domain restriction.
Another form of the idea shows up in restrictions placed on the domain of a vector-valued function simply because only part of that domain is interesting in the application at hand. This often shows up in connection with plotting, for example one cycle of a helix might be defined by h(t) = ⟨cos t, sin t, t⟩ , for 0 ≤ t ≤ 2π.
Use the ParametricPlot3D function, which looks like
ParametricPlot3D[ { x, y, z }, {var, low, high} ]
where x, y, and z are expressions involving some variable for x, y, and z coordinates along the curve to plot, var is that variable, and low and high are the low and high bounds on its value. This function plots a 3-dimensional vector-valued function; an analogous ParametricPlot function plots 2-dimensional ones.
Download this notebook for examples of both 2- and 3-dimensional vector-valued functions and their plots.
Start doing some calculus with vector-valued functions, particularly finding limits of them.
Please read “Limits and Continuity of a Vector-Valued Function” in section 2.1 of the textbook.