Problem Set 3 — Lines and Planes

Purpose

This exercise develops your ability to work with and think about lines and planes in three dimensions. In the process, it also gives you some practice working with dot and cross products of vectors. It therefore contributes to the following learning outcomes for this course:

• Outcome 1. Represent vectors analytically and geometrically, and perform vector calculations
• Outcome 2. Use vector equations to represent lines and planes.

Background

This exercise is based on material in sections 1.3 through 1.5 of our textbook. We talked about those sections in classes between September 12 and 16.

Activity

Solve each of the following problems.

Problem 1

Spacey the Space Traveler is playing zero-gravity space baseball. Spacey hits the ball when it is at point \((1,1,-2)\), and sends it moving in direction \(\langle -1, 10, 4 \rangle\). (If you care, I’m thinking of these things relative to Spacey’s right-forward-up coordinate system, with distances measured in feet.) Because there is no gravity or air friction in zero-gravity space baseball, the ball will keep moving in this direction forever, unless one of the other players catches it.

Part A

Give an equation for the line along which the ball moves.

Part B

The center of the centerfielder’s glove is at point \((-49, 501, 200)\), and doesn’t move. Does the centerfielder catch Spacey’s ball?

Problem 2

Find a scalar equation for the plane that contains point \(P(3,2,2)\) and has normal vector \(\vec{n} = \langle 2, 3, -1 \rangle\).

Problem 3

Imagine a plane that contains points \(P(1,3,-1)\) and \(Q(0,-7,2)\). Furthermore suppose that vector \(\vec{v} = \langle -3, 0, -1 \rangle\) is perpendicular to the plane. Find a third point, not colinear with (i.e., not in the same line as) \(P\) and \(Q\), that is also in the plane. (Note: there are many ways to solve this problem.)

Problem 4

One elegant way of doing computer graphics is called “ray tracing,” because it traces (in reverse) rays of light that reach a virtual eye from some virtual scene. In other words, it traces rays of light backwards from the eye into the scene, to see what objects they came from and therefore what color the light was.

Suppose one of these virtual rays of light is directed along the line \(\vec{r}(t) = (0,-2,0) + t \langle 0, 2, 2 \rangle\). The plane \(x - y + 2z - 10 = 0\) represents the side of a hill in the virtual world. Where does the ray intersect the hill?

Problem 5

Describe a way you could determine whether an arbitrary line, \(\vec{r}(t) = P + t\vec{v}\), is parallel to an arbitrary plane, \(ax + by + cz + d = 0\). Demonstrate your method by using it to decide whether line \(\vec{r}(t) = (0, 1, 0) + t\langle 1, 1, 1 \rangle\) is parallel to the plane \(x + y - 3 = 0\).

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. 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.