Problem Set 3—Lines and Planes

Purpose

This problem set reinforces your understanding of linear objects in 3-dimensional space and of how they can be defined and manipulated using vectors.

Background

This problem set is based on material in section 12.5 of our textbook. We will discuss this material in class between September 18 and September 21.

Activity

Solve each of the following problems:

Problem 1

Problem 2 in section 12.5 of our textbook (give parametric equations for the line through points (1,2,-1) and (-1,0,1)).

Problem 2

Problem 64 in section 12.5 of our textbook (give two equations for a plane, one based on (4,1,5) being a point in the plane and i - 2j + k being a normal to it, and the other based on (3,-2,0) being a point in the plane and -√2i + 2√2j - √2k being a normal).

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

Is it possible to have a line in 4-dimensional space? Explain why or why not, and if it is possible, give parametric equations for the line that passes through (1,0,1,2) with direction ⟨0,-1,3,1⟩.

Problem 5

Problem 68 in section 12.5 of our textbook (explain how to determine whether two plans are parallel; how to determine if two planes are perpendicular).

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.