Problem Set 3 — Vectors

Purpose

This exercise reinforces your understanding of vectors and common vector operations. It therefore contributes to the following learning outcome for this course:

• Outcome 1. Represent vectors analytically and geometrically, and perform vector calculations

Background

This exercise is based on material in sections 1.2 through 1.4 of our textbook. We discussed this material in classes between February 6 and 10.

Activity

Solve each of the following problems.

Problem 1

Do each of the following vector calculations.

1. \(\langle 2, 5, -1 \rangle + 3 \langle -5, 4, 4 \rangle\)
2. \(\langle 1, 5, 3 \rangle \cdot \langle 6, -4, 4 \rangle\)
3. \(\frac{-2}{3}\langle 6, -9, -3\rangle \cdot \langle 4, 1, 0 \rangle\)
4. \(\langle 1, -2, 1 \rangle \times \langle 0, 2, -1 \rangle\)
5. \(\langle -2, 1, -1 \rangle \cdot \left(\langle -2, 1, 1 \rangle \times \langle 3, -1, 2 \rangle\right)\)
6. \(\langle 1, 2, 3, 4 \rangle - 4 \langle 3, -2, 1, -1 \rangle\)

Problem 2

Let vector \(\vec{p}\) be the projection of \(\langle 3, 1, -2 \rangle\) onto \(\langle 0, 1, 1 \rangle\), and let \(\vec{u}\) be another vector such that \(\langle 3, 1, -2 \rangle = \vec{p} + \vec{u}\).

Part A

Find the projection, \(\vec{p}\).

Part B

Find the second vector, \(\vec{u}\).

Part C

Show that \(\vec{p}\) and \(\vec{u}\) are perpendicular to each other.

Problem 3

A function, \(f(x)\), is said to be linear if the following hold, for all scalar constants \(a\) and values \(x\) and \(y\):

• \(f(ax) = a f(x)\)
• \(f(x + y) = f(x) + f(y)\)

So informally, you can think of linear functions as ones that you can factor multiplication out of, and distribute over addition. For example, the function that multiplies a vector by 2, \(f(\vec{v}) = 2\vec{v}\), is linear because \(f(a\vec{v}) = 2(a\vec{v}) = (a2)\vec{v} = a(2\vec{v}) = a f(\vec{v})\), and \(f(\vec{v} + \vec{u}) = 2(\vec{v} + \vec{u}) = 2\vec{v} + 2\vec{u} = f(\vec{v}) + f(\vec{u})\).

Is the magnitude operation on vectors linear? Either explain why it meets the two requirements above, or show an example of vectors and/or scalars that violate one or both of the requirements.

Problem 4

Imagine that Geneseo Video Games Studio is a start-up business that develops video games. Their very first game is a first-person exploring/adventure game. As we discussed in class, they need to find vectors that point in the directions the player sees as “right,” “up,” and “backwards” as they move around the virtual world in order to draw that world as seen by the player. Suppose the programmers at Geneseo Video Games Studio have figured out how to find the “right” and “up” vectors, but are stuck on the “backwards” one. They hire you as a consultant to advise them. Explain a way to generate a “backwards” vector from “right” and “up” vectors. Illustrate your method by applying it to “right” vector \(\langle 3, 2, -2 \rangle\) and “up” vector \(\langle 0, 1, 3 \rangle\).

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.