Problem Set 4 — Lines and Planes

Purpose

This exercise reinforces your understanding of representations of lines and planes in space, and how to use those representations to reason about lines and planes. 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 mainly based on material in section 1.5 of our textbook. We covered that material in classes between February 13 and 17. Many of the operations on lines and planes involve vector calculations of the sort covered in sections 1.2 through 1.4 of the textbook and discussed in classes between February 6 and 10.

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?

Part C

At exactly the moment Spacey hits the ball, Spot the Space Dog starts chasing it from point \((101, 501, -102)\). Spot moves along the line \(\vec{s}(t) = (101, 501, -102) + t \langle -2, 5, 5 \rangle\). Assuming that the direction vectors for the ball and Spot give their actual velocities in feet per second, does Spot catch the ball? In other words, are Spot and the ball ever at the same place at the same time? (Hint: notice that this is a slightly different question than superficially similar ones we looked at in class about whether two lines intersect.)

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, and should ordinarily last half an hour. 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.