Problem Set 4—Lines, Planes, and Curves

Purpose

This problem set reinforces your understanding of lines, planes, and curves in 3 dimensions.

Background

This problem set is based on material in sections 12.5 and 13.1 of our textbook. We discussed, or will discuss, this material in class between February 4 and 11.

Activity

Solve each of the following problems:

Problem 1

Exercise 66 in section 12.5 of our textbook (find equations defining the line in the z = 3 plane that makes angles of π/6 radians and π/3 radians respectively with i and j).

Problem 2

Find the point at which the lines ⟨ 1+3t, -2t ⟩ and ⟨ 3-s, 2+s ⟩ intersect.

Problem 3

Problem 32 in section 12.5 of our textbook (find an equation for the plane containing points (1,2,3) and (3,2,1) and perpendicular to plane 4x - y + 2z = 7).

Problem 4

Exercise 15 in the “Additional and Advanced Exercises” for chapter 12 in our textbook (find the projection of ⟨ 1, 1, 1 ⟩ onto the plane x + 2y + 6z = 6; see book for details).

Problem 5

Exercise 18 in section 13.1 of our textbook (find the angle between a particle’s velocity and acceleration when t = 0 if its position is given by r(t) = ⟨ (4/9)(1+t)^3/2, (4/9)(1-t)^3/2, (1/3)t ⟩).

Problem 6

Exercise 20 in section 13.1 of our textbook. Also use muPad to plot r(t) and your tangent line to visually check your work. (The question asks you to find the tangent line to r(t) = ⟨ t², 2t-1, t³ ⟩ at t = 2.)

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.