Unformatted text preview: APPM 2350 HOMEWORK 02 FALL 2010 1. Suppose you are the Captain of the USS Gauss, a brand new spacecraft, and you have just been launched into space to ﬂy to the moon. Your helper monkey Mojo has determined that the best landing spot on the moon has coordinates (1, −2, 3). You are currently at the point (−5, 3, 4) and ﬂying in the direction 2 i − j. (a) How far are you from the landing spot? (b) What is the parametric equation describing your ﬂight path? (c) Will you hit your landing spot? If not, how close will you get? (d) What path do you need to take to ensure that you hit the landing spot? 2. After successfully landing the USS Gauss on the moon, you begin your return trip home. You and Mojo are enjoying some freeze dried ice cream when suddenly, Mojo sees an astroid out the window. The astroid is located at the point (0,5,-1) and has a heading of 2 i + 1 j + 3 k. Your current ﬂight path is the line described by r(t) = (−1 + t) i + (2 − t) j. Will you hit the astroid? If not, how close will you get? (Hint: if two lines do not intersect, they can be embedded in parallel planes.) 3. Consider a room ten units long in the x, y , and z directions. Speciﬁcally, the walls of the room are the four planes, x = 0, x = 10, y = 0, and y = 10, and the ﬂoor and ceiling are z = 0 and z = 10, respectively. A ﬂat triangular mirror is mounted in one of the corners of the ceiling. The corners of the mirror are at locations (10, 9, 10), (10, 10, 9), and (9, 10, 10). You are sitting at location (5, 0, 0) playing with your new green laser pointer. (a) If you aim your laser pointer directly at the corner of the room with coordinates (10, 10, 10), determine the coordinates where the beam will hit the walls, or ﬂoor, of the room. (Hint: an incoming ray of light, and the surface normal where the ray hits the surface, form a plane. The reﬂected ray is in the same plane. Also, the angle between the incoming ray and the normal is the same as the angle between the normal and the reﬂected ray.) (b) Suppose the ﬂat mirror is replaced with one octant of a spherical mirror of radius 1. The corners of the new spherical mirror are again located at (10, 9, 10), (10, 10, 9), and (9, 10, 10). If you again aim your laser pointer directly at the corner of the room with coordinates (10, 10, 10), determine the new coordinates where the beam will hit the walls, or ﬂoor, of the room. (Hint: a sphere has the property that at a point P on the surface, the normal to the surface, n, is parallel to the vector from the center of the sphere to the point P .) 4. Suppose you have the quadric surface Ax2 + By 2 + Cz 2 + F xz + K = 0. (a) What restrictions must you place on the coeﬃcients of your quadric surface to ensure the surface passes through the origin? (b) What restrictions must you place on the coeﬃcients of your quadric surface to ensure the surface passes through the y-axis? (c) Assume that we label the octants of (x, y, z ) space in the following way: Octant I : x > 0, y > 0, z > 0 Octant II : x > 0, y < 0, z > 0 Octant III : x < 0, y < 0, z > 0 Octant IV : x < 0, y > 0, z > 0 Octant V : x > 0, y > 0, z < 0 Octant V I : x > 0, y < 0, z < 0 Octant V II : x < 0, y < 0, z < 0 Octant V III : x < 0, y > 0, z < 0 Suppose you a have graph of your quadric surface in Octant I . In which other octants do you immediately know the shape of your surface? Now, suppose you have a graph of your surface in Octant III . In which other octants do you know the shape of your surface? (d) Suppose your surface passes through the origin. What conditions must be imposed on the coeﬃcients so that z = x, y = 0, is a line on the surface? If this line is on the surface, can the line z = −x, y = 0 also be on the surface? Explain why or why not. (e) Now suppose your surface passes through the origin and intersects the line z = x, y = 0, and ﬁnally that A = C . You introduce the following change of variables x = z = x z √ −√ 2 2 x z √ +√ . 2 2 (f) Transform your expression for the quadric surface into a quadric surface in the variables (x , y, z ). Simplify the expression for your new quadric surface as much as possible. (g) Given your new expression for the quadric surface, what conditions must you enforce on your coeﬃcients such that you will have a surface at all? Assuming thesse conditions, draw your surface in the (x , y, z ) coordinate system. Then using your insight from above, draw your surface in the (x, y, z ) coordinate system. Solve for x and z in terms of x and z . How are the x and z axes related to the x and z axes? (Hint: What do the lines z = x and z = −x become in the new coordinates?) ...
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This note was uploaded on 02/10/2011 for the course APPM 2350 taught by Professor Adamnorris during the Fall '07 term at Colorado.
- Fall '07