Problem Set 3—Vector Products

Purpose

This problem set consolidates your understanding of vector dot and cross products.

Background

This problem set is based on material in sections 12.3 and 12.4 of our textbook. We discussed, or will discuss, this material in class on February 2 and 3.

Activity

Solve each of the following problems:

Problem 1

Exercise 2 in section 12.3 of our textbook (find the dot product v⋅u given v = ⟨ 3/5, 0, 4/5 ⟩ and u = ⟨ 5, 12, 0 ⟩; also find the lengths of both vectors, the cosine of the angle between them, the scalar component of u in the direction of v, and the projection of u onto v).

Problem 2

A slight clarification of exercise 8 in section 12.4 of our textbook: find the cross products u×v and v×u, and their lengths, given u = ⟨ 3/2, -1/2, 1 ⟩ and v = ⟨ 1, 1, 2 ⟩.

Problem 3

Exercise 26 in our textbook’s practice exercises for chapter 12 (page 734; find the area of the parallelogram bounded by ⟨ 1, 1, 0 ⟩ and ⟨ 0, 1, 0 ⟩; then find the volume of the parallelpiped with those 2 sides and third side ⟨ 1, 1, 1 ⟩).

Problem 4

A virtual character in a video game is facing in direction ⟨ 2, 1, 2 ⟩ in the game’s coordinate system. Find vectors (they don’t have to be unit vectors) that point in the directions this character perceives as “up” and “right.”

Problem 5

Exercise 18 in section 12.3 of our textbook (show that vectors along certain chords of a circle are orthogonal; see book for details).

Problem 6

Exercise 28h in section 12.4 of our textbook (is it true that (u×v)⋅u = v⋅(u×v)).

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.