Problem Set 2 — Introduction to Vectors

Purpose

This exercise introduces you to some basic ideas, applications, and calculations related to vectors. It also gives you some additional practice plotting 3-dimensional surfaces. 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 based on material in the “Working with Vectors in R³” part of section 1.2 of our textbook, as well as ideas about quadric surfaces from section 1.6, and plotting with Mathematica from class. We discussed these things in classes between August 31 and September 9.

Activity

Solve each of the following problems.

Problem 1

It is a little-known, and even less-believed, “fact” that the Bronze Bear statue in Geneseo marks the origin of the universe’s coordinate system. The universe’s positive \(x\) axis runs south down the center of Main Street, the positive \(y\) axis runs east along Center Street, and the positive \(z\) axis runs up through the center of the statue.

Part A

Suppose you are pedaling a bicycle north along Main Street in such a manner that you create a force of 2 pounds pushing the bicycle forward. Describe this force as a vector (1) in component form, and (2) in terms of standard unit vectors \(\mathbf{\hat{i}}\), \(\mathbf{\hat{j}}\), and \(\mathbf{\hat{k}}\).

Part B

Now suppose a wind kicks up, exerting force \(\langle 1, 1, 0.2 \rangle\) (in pounds) on you. Describe the direction of this force in English. What is the net force acting on you (assume you keep pedaling as in Part A).

Part C

If the wind gets stronger so that it exerts twice the force it did in Part B, while still blowing in the same direction, what is the new net force on you?

Part D

Every so often, cars or trucks collide with the bear fountain, and damage it (this part is even true). While the bear is being repaired after these accidents, the Universe Distances and Coordinates Authority temporarily relocates the origin of the universe to an unused salt mine west of Geneseo (this part may not be as true). The temporary origin is 3 miles west of the bear fountain, half a mile south, and half a mile below it. The temporary axes point in the same directions as the original ones do. Describe the relocation as a displacement vector, i.e., a vector whose components are the origin’s change in \(x\), \(y\), and \(z\) coordinates.

Part E

Let \((x^\prime, y^\prime, z^\prime)\) denote the coordinates in the temporary universe coordinate system of the point whose coordinates in the original system are \((x,y,z)\). Give an equation involving the displacement vector from Part D that lets you calculate \((x^\prime, y^\prime, z^\prime)\) from \((x,y,z)\).

Problem 2

In section 1.1, our textbook asserts a number of algebraic properties of 2-dimensional vector arithmetic (for example, that vector addition is commutative, that scalar multiplication distributes over vector addition, etc.) The book proves some of these properties and leaves other as exercises for the reader. There’s no similar list of properties for 3-dimensional vectors; you’re supposed to believe that the same properties apply to 3 dimensions as to 2 (as indeed they do; in fact they apply to any number of dimensions).

One of the properties asserted without proof is that scalar multiplication is associative, i.e., that if \(r\) and \(s\) are scalars and \(\vec{u}\) is a vector, then \(r(s\vec{u}) = (rs)\vec{u}\). Prove this for 3-dimensional vectors. (Hint: the “Proof of Commutative Property” and “Proof of Distributive Property” boxes in section 1.1 give examples of similar proofs.)

Problem 3

For each of the following equations, try to determine what sort of quadric surface it describes without plotting that surface, including any details you can about what axis (if any) the surface aligns with, where its center is, etc. Then plot the surface with Mathematica to check your determination.

(Note: I will want to see your Mathematica work when we meet to grade this problem set. There’s no way I can tell whether you really did it after rather than before deciding what sort of surface you were looking at, and I won’t try to. But I probably will ask you to solve a similar problem during grading, when obviously I can see that you’re deducing things about the surface before you look at the plot, so getting the practice while working on this problem will be helpful.)

1. \(x^2 - y^2 + z^2 = 4\)
2. \(z = \frac{y^2}{4} - \frac{x^2}{9}\)

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.