Problem Set 1—3D Coordinates

Purpose

This problem set reinforces your understanding of 3-dimensional coordinate systems. The key concepts it exercises are describing regions of space with 3D coordinate systems, plotting with muPad, and distance in 3 dimensions.

Background

This problem set is based on material in section 12.1 of our textbook. We discussed this section in class on January 20. This problem set also draws on material about muPad (the computer algebra system in Matlab) that is not in the text, but that was discussed in class on January 21.

Activity

Solve each of the following problems:

Problem 1

We looked in class at how the equation z = y² defines a sort of parabolic tube in 3 dimensions. Give inequalities that describe all the points on or below this tube in the first octant (notice that the first octant is technically defined as points satisfying x ≥ 0, y ≥ 0, and z ≥ 0).

Problem 2

Define the “egg carton function” by the equation z = cos(x)sin(y). Use muPad to plot the egg carton function over a suitable range of x and y values to understand why I called it the egg carton function.

Problem 3

Use a muPad plot to determine whether the plane x + y - z = 2 passes above or below the origin. What do you mean by “above” and “below” in this context?

Problem 4

Exercise 62 in section 12.1 of our textbook (show that (3,1,2) is equidistant from (2,-1,3) and (4,3,1)).

Problem 5

Exercise 64 in section 12.1 of our textbook (find an equation for the set of points equidistant from (0,0,2) and the xy plane).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.