Problem Set 1 — 3-Dimensional Coordinates

Purpose

This exercise reinforces your understanding of 3-dimensional coordinates and some basic calculations related to them. As such, it contributes to the following learning outcomes for this course:

• Outcome 1. Represent vectors analytically and geometrically, and perform vector calculations
• Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.

Background

This exercise is mostly based on material in the first part of section 1.2 of our textbook. We discussed that material in class on January 27.

This exercise also asks you to use Mathematica to plot some equations in 3 dimensions. We talked (or will talk) about how to do such plotting in class on January 29.

Activity

Solve each of the following problems.

Problem 1

Remember the mouse, cheese, and cat video we watched on the first day of class.

Part A

At the beginning of the video, the mouse knocks a piece of cheese out of a mousetrap. Suppose that the video’s virtual world is defined by a coordinate system centered on the mousetrap, with axes parallel to the walls and floor of the surrounding room, and distances measured in centimeters (I’m sure that the virtual world was defined relative to some coordinate system, but this specific one is completely my own invention). Suppose that in this coordinate system, the mouse is located at point $(-2,0,0)$, the cheese lands at $(100,-10,0)$, and the cat is located at $(110,95,1)$. Who is closer to the cheese, the cat or the mouse? What are the distances?

Use Mathematica as a calculator to do the arithmetic in this question.

Part B

Towards the end of the video, the cat swishes its tail into the mousetrap. Suppose that the tail is 20 centimeters long, and can swing a full 360 degrees around its base. There is thus a sphere of radius 20 cm around the cat’s rear end in which mousetraps are a danger to the cat. Supposing that the base of the cat’s tail is at point $(-10,0,0)$, give the equation for this sphere.

Part C

Use Mathematica to plot the sphere whose equation you derived in Part B.

Problem 2

What shape(s) will the equation $x^2-3x+2=0$ produce when plotted in 3 dimensions? When you think you have an answer, plot the equation with Mathematica to check it. Your conjectured answer should help you decide what ranges of $x$, $y$, and $z$ values to plot over. Be prepared to explain in English when you meet with me to go over this problem set why the equation produces the shape(s) it does.

Problem 3

Renowned time traveler Dr. Whowhatwhenwherewhyandhow travels from spacetime point $(1,3,2,6)$ to $(-1,5,4,4)$. How far has the Doctor traveled? (Assume time travelers measure space and time in units of mileyears — being merely 3-dimensional creatures, we can’t really visualize what a mileyear looks like, but somehow it is a unit that manages to apply equally to “distance” in space and time. So, for example, you can interpret the Doctor’s starting point as being $1$ mileyear from the origin in the $x$ direction, $3$ mileyears in the $y$, $2$ mileyears in the $z$, and $6$ in the time direction; the question is asking how many mileyears the Doctor has traveled.)

Problem 4

Our textbook leaves the proof of the 3-dimensional distance formula as an exercise (although I don’t actually see it among the exercises for section 1.2). Give a proof, particularly giving the details of how the Pythagorean Theorem fits in. You may find Figure 1.2.8 in the book, and its terminology, helpful.

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above, and should ordinarily last half an hour. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.