Problem Set 2—Quadrics and Vectors

Purpose

This problem set reinforces your understanding of quadric surfaces and vectors.

Background

This problem set is based on material in sections 12.2 and 12.6 of our textbook. We discussed quadric surfaces (section 12.6) in class on January 26, and vectors (section 12.2) on January 27 and 28.

Activity

Solve each of the following problems:

Problem 1

Exercise 70 in the “Practice Exercises” for chapter 12 of our textbook, using muPad to sketch the surface (identify and graph the surface defined by by y = -(x²+z²)).

Problem 2

Exercise 6 in section 12.2 of our textbook (if u = ⟨3,-2⟩ and v = ⟨-2,5⟩, express -2u + 5v in component form and find its magnitude).

Problem 3

A hiker trying to get around a swamp walks the following path:

• First, she walks √2 miles at a heading of 45° (headings are measured clockwise from due north)
• Then she turns to a heading of 0° and walks 3 miles
• Next she turns to a heading of -90° and walks 1 mile
• Finally she turns to a heading of -45° and walks √2 more miles.

Part A

Where does the hiker end up relative to her original position?

Part B

How far is she, in a straight line distance, from her original position?

Problem 4

Exercise 42 in section 12.2 of our textbook (decompose ⟨1,-2⟩ into a sum of two other vectors, one of which is parallel to ⟨2,3⟩ and the other of which is parallel to ⟨1,1⟩).

Problem 5

What is the magnitude of ⟨-1,2,-2,4⟩ + 2⟨1,1,2,-3⟩?

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.