MATH 2203 Exam 1 January 26, 2004 S. F. Ellermeyer Name

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1 MATH 2203 Exam 1 January 26, 2004 S. F. Ellermeyer Name Instructions. This exam contains seven problems, but only six of them will be graded. You maychooseanysixtodo. PleasewriteDON TGRADEontheonethatyoudon twantme to grade. In writing your solution to each problem, include sufficient detail and use correct notation. (For instance, don t forget to write = when you mean to say that two things are equal.) Your method of solving the problem should be clear to the reader (me). If I have to struggle to understand what you have written, then you might not get full credit for your solution even if you get a correct answer. 1. One of the diameters of a sphere has endpoints ( 2, 6, 4) and (, 2, 9). (a) Find the radius of this sphere. The diameter of the sphere is q ( ( 2)) 2 +(2 6) 2 +(9 4) 2 = 90 so the radius is 90/2. (b) Find the center of this sphere. The center of the sphere is at the point µ 2+, , 4+9 = 2 µ 3 2, 4, (c) Write an equation for this sphere. An equation for this sphere is µ x 3 2 µ +(y 4) z 13 2 = (a) Let a and b be the vectors a= h2, 1i b= h8, 12i. Draw pictures of the standard representatives of a and b. 1

2 (b) Find the vector u =proj a b and the vector v = b u. u = proj a b = b a a 2 a = 28 h2, 1i 6 =, 28 À and 6 v = b u = h8, 12i, 28 À = 16, 32 À. (c) Show that u + v = b and that u is orthogonal to v. Illustrate this result including the vectors u and v in the drawing that you made in part a. 6 u + v =, 28 À + 16, 32 À = h8, 12i = b and u v = µ µ µ 28 µ 32 =0 so u and v are orthogonal (because their dot product is zero). (a) A child pulls a sled through the snow with a force of 0 Newtons exerted at an angle of 30 above the horizontal. Find the horizontal and vertical components of this force. The force vector can be written as F =0cos(30 ) i +0cos(60 ) j 2

3 so the horizontal component of the force is 30 cos (30 )=2 3 Newtons and the vertical component of the force is 30 cos (60 )=2Newtons. (b) Assuming that the child pulls the sled in this way a distance of 300 meters, how much work is done (in Joules)? The displacement vector is D = 300i sotheamountofworkdonebytheforceis ³ D F = , 990 Joules. (c) If a and b are the vectors pictured below, does the vector a b point into the page or out of the page? How do you know? By the right hand rule, the vector a b points into the page. (d) If we were to make vector a be longer (but not change the length of vector b and not change the angle between a and b), what effect would that have on the vector a b. Inparticular,woulda b become shorter or longer? Would a b point into the page or out of the page? How do you know? The vector a b would still point into the page by the right hand rule. Also, since a b = a b sin (θ), the length of the vector a b would increase. (e) Ifweweretoincreasetheanglebetweena and b (but still keep a and b the same length and still keep the angle acute), what effect would that have on the vector a b? In particular, would a b become shorter or longer? Would a b point into the page or out of the page? How do you know? By the right hand rule, the vector a b would still point into the page. However, since sin (θ) would increase, the length of the vector a b would increase. (f)ifweweretoincreasetheanglebetweena and b to be just slightly less than 180 (but still keep a and b thesamelength),whateffect would that have on the vector a b? Inparticular,woulda b becomeshorterorlonger? Woulda b point into the page or out of the page? How do you know? 3

4 The vector a b would still point into the page by the right hand rule. The length of a b would decrease, however, since sin (θ) would be very small. Find the point at which the line intersects the plane x =1 t y = t z =1+3t 4x 2y z =6. The line intersects the plane at the parameter value (t) such that 4(1 t) 2(t) (1 + 3t) =6. The value of t for which the above equation is true is t = 1/3. Thus the line and the plane intersect at the point 4 3, 1 3, Find parametric and/or symmetric equations for the line of intersection of the two planes x 6y 2z =12 3x +8z =0. Theequationsoftheplanescanberewrittenas 1x +18y +6z = 36 1x +40z =0. Thus, any point lying in both planes must satisfy both equations 9y +23z =0 3x +8z =0. Two such points are (0, 2, 0) and (8/3, /9, 1). Thus, a vector parallel to the line of intersection is 8 v = 3, 23 À 9, 1. Parametric equations for this line of intersection are x = 8 3 t y = t z = t. Symmetric equations for this line are x 8 3 = y = z 1. 4

5 3. Here are six equations labelled 1 6 and six surfaces labelled A F. Match each equation with the surface that it describes.

6 1) x 2 + y 2 z 2 =1 2) x 2 y 2 z =0 3) x 2 + y 2 z =0 4) x 2 + y 2 z 2 = 1 ) x 2 + y 2 z 2 =0 6) x 2 + y 2 + z 2 =1 Equation 1 matches surface B. Equation 2 matches surface C. Equation 3 matches surface F. Equation 4 matches surface E. Equation matches surface D. Equation 6 matches surface A. 6

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)