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 August 31.

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 September 2.

Finally, some of the problems in this exercise are phrased in terms of “vectors.” We’ll talk in detail about vectors in about a week. For now, think of a vector as a line segment with a direction — in other words, a line segment that starts at some point and ends at another, or, equivalently, that runs from one point to another. If you want to read ahead a bit, the second half of section 1.2 in the book introduces 3-dimensional vectors, but you shouldn’t need that material for this problem set.

Activity

Solve each of the following problems.

Problem 1

Use Mathematica to plot the equation \(x^2 - 1 = 0\) over the 3-dimensional region \(-2 \le x \le 2, -2 \le y \le 2, -2 \le z \le 2\). Be prepared to explain in English why this equation produces the surface(s) you see in Mathematica.

Problem 2

Remember the computer animated snake from our introductory class? Now that we understand 3-dimensional coordinate systems, we can start talking about positions of parts of it. For example, suppose the tip of the snake’s tail is at point \((-10,1,0)\), and that its body twists around in the \(z = 0\) plane to the origin. At the origin, the snake’s neck abruptly goes straight up for \(2\) units, to the center of the snake’s head.

Part A

What are the coordinates of the center of the snake’s head?

Part B

Imagine a vector from the tip of the snake’s tail to the center of its head. How long is that vector?

Part C

Suppose the snake’s head is a sphere \(1\) unit in radius. Give an equation for the head. Then plot that equation with Mathematica to verify that it is indeed a sphere in the expected place (hint: pick a region to plot over that will make it easy to see the whole shape and where it is).

Problem 3

Renowned time traveler Dr. Whowhatwhenwherewhyandhow travels from spacetime point \((1,3,2,6)\) to \((-1,5,4,4)\). To put it another way, he travels along the vector from \((1,3,2,6)\) 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. 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.