(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 225: Calculus III Assignment 2 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Name: Solution :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Signature: :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Instructions :[font = text; inactive; preserveAspect] This assignment contains problems on vectors, dot and cross products, and lines and planes in space, from Chapter 11 in Finney & Thomas and Chapter 1 in the Lectuire Notes. You should look in AdvanceCalculusDemo and Exercises in the Notebook folder or the back of the Lecture Notes as well as the the early quiz notebooks in the Quizzes folder for examples of how to do the following problems. The functions Norm[], Cross[], and Pr[] are explained in AdvanceCalculusDemo. Problem 4 can be solved using cross products. You should use the Solve[] command in Problem 5. Don't forget to Clear[] or Remove[] variables before moving on to the next problem. Be sure to type in comments explaining what you are doing. It is best to change the style of such comments to Text. To do this, select the cell by clicking on the bar at the right and select Text from the pop-up menu in the ruler above (or hit COMMAND-7). Remember to uncheck Show In/Out Names in the File menu and close this group before printing. ;[s] 27:0,0;192,1;211,0;216,1;225,0;408,2;414,0;416,2;423,0;429,2;433,0;451,1;470,0;539,2;546,0;586,2;593,0;597,2;605,0;765,1;769,0;846,1;850,0;932,1;949,0;957,1;961,0;1005,-1; 3:14,13,9,Times,0,12,0,0,0;7,13,9,Times,1,12,0,0,0;6,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] *) <1-t, y->1+t, z->3t} :[font = output; output; inactive; preserveAspect; endGroup] {1 - t, 1 + t, 3*t} ;[o] {1 - t, 1 + t, 3 t} :[font = text; inactive; preserveAspect] The squared distance from p to q is :[font = input; preserveAspect; startGroup] f[t_] = (p-q).(p-q) :[font = output; output; inactive; preserveAspect; endGroup] (-1 - t)^2 + t^2 + (-3 + 3*t)^2 ;[o] 2 2 2 (-1 - t) + t + (-3 + 3 t) :[font = input; preserveAspect; startGroup] f'[t] :[font = output; output; inactive; preserveAspect; endGroup] -2*(-1 - t) + 2*t + 6*(-3 + 3*t) ;[o] -2 (-1 - t) + 2 t + 6 (-3 + 3 t) :[font = input; preserveAspect; startGroup] Solve[f'[t] == 0,t] :[font = output; output; inactive; preserveAspect; endGroup] {{t -> 8/11}} ;[o] 8 {{t -> --}} 11 :[font = text; inactive; preserveAspect] The corresponding point is :[font = input; preserveAspect; startGroup] p /. {t->8/11} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {3/11, 19/11, 24/11} ;[o] 3 19 24 {--, --, --} 11 11 11 :[font = input; preserveAspect] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 4 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] a) Find an equation for the plane through the points p = {1,0,-2}, q = {0,-3,1}, and r = {-3,1,0}. b) Find the parametric equations for the line in which the planes x - y + z == 1 and x + y + 3z == -2 intersect. ;[s] 14:0,2;4,0;54,1;66,0;72,1;84,0;90,1;102,0;104,2;108,0;171,1;185,0;194,1;210,0;222,-1; 3:7,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Solution :[font = text; inactive; preserveAspect; startGroup] a) Use the cross product of two vectors parallel to the plane to get a vector perpendicular. ;[s] 2:0,1;2,0;93,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] p = {1,0,-2}; q = {0,-3,1}; r = {-3,1,0}; ;[s] 2:0,0;40,1;42,-1; 2:1,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] v1 = q-p :[font = output; output; inactive; preserveAspect; endGroup] {-1, -3, 3} ;[o] {-1, -3, 3} :[font = input; preserveAspect; startGroup] v2 = r-p :[font = output; output; inactive; preserveAspect; endGroup] {-4, 1, 2} ;[o] {-4, 1, 2} :[font = input; preserveAspect; startGroup] n = Cross[v1,v2] :[font = output; output; inactive; preserveAspect; endGroup] {-9, -10, -13} ;[o] {-9, -10, -13} :[font = text; inactive; preserveAspect] The equation of the plane is :[font = input; preserveAspect; startGroup] n.{x,y,z} == n.p :[font = output; output; inactive; preserveAspect; endGroup] -9*x - 10*y - 13*z == 17 ;[o] -9 x - 10 y - 13 z == 17 :[font = text; inactive; preserveAspect] b) To find the inetersection we solve the equations simultaneously, say, solve for x and y in terms of z. We then get a parameterization of the line by substituting z == t. ;[s] 10:0,1;2,0;83,2;84,0;89,2;90,0;103,2;104,0;165,2;171,0;173,-1; 3:5,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[{x - y + z == 1, x + y + 3z == -2},{x,y}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 1 + (-3 - 2*z)/2 - z, y -> (-3 - 2*z)/2}} ;[o] -3 - 2 z -3 - 2 z {{x -> 1 + -------- - z, y -> --------}} 2 2 :[font = input; preserveAspect; startGroup] Simplify[%] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -1/2 - 2*z, y -> -3/2 - z}} ;[o] 1 3 {{x -> -(-) - 2 z, y -> -(-) - z}} 2 2 :[font = text; inactive; preserveAspect] The parametric equations of the line are: x == -1/2 - 2 t y == -3/2 - t, z == t ;[s] 3:0,0;42,1;91,0;92,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] Another way to solve this problem is to notice that the normal vectors of the two planes are both perpendicular to the line, so their cross product will be parallel to the line, i.e. give the direction vector of the line. :[font = input; preserveAspect] n1 = {1,-1,1}; n2 = {1,1,3}; :[font = input; preserveAspect; startGroup] v = Cross[n1,n2] :[font = output; output; inactive; preserveAspect; endGroup] {-4, -2, 2} ;[o] {-4, -2, 2} :[font = text; inactive; preserveAspect] Two get a point p on the line, find any simultaneous solution to the equations. For example, set z == 0, then the equations simplify to: ;[s] 5:0,0;16,1;17,0;97,1;103,0;137,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] x - y == 1; x + y == -2; ;[s] 3:0,0;13,1;14,0;28,-1; 2:2,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect] which gives (add and subtract the equations): :[font = input; preserveAspect] x == -1/2; y == -3/2; :[font = input; preserveAspect] p = {-1/2,-3/2,0}; :[font = text; inactive; preserveAspect] The parametric equations of the line are then :[font = input; preserveAspect; startGroup] p + t v :[font = output; output; inactive; preserveAspect; endGroup] {-1/2 - 4*t, -3/2 - 2*t, 2*t} ;[o] 1 3 {-(-) - 4 t, -(-) - 2 t, 2 t} 2 2 :[font = text; inactive; preserveAspect; endGroup] or x == -1/2 - 4t, y == -3/2 - 2t, and z == 2t. This looks different then the previous solution, but it does represent the same line. The direction vector from these equations is {-4,-2,2} which is parallel to the direction vector {-2,-1,1} from the previoius solution. ;[s] 11:0,0;2,1;17,0;19,1;33,0;39,1;46,0;179,1;188,0;231,1;240,0;270,-1; 2:6,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 5 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] a) Find the point in which the line x == t, y == -t, z == -4t meets the plane x - 2y + z == 6. b) Find the point in which the three planes x - 2y + z == 6, 2x + y - z == 2, and y + 3z == 4 intersect. ;[s] 16:0,2;4,0;37,1;62,0;83,1;98,0;100,2;102,0;145,1;160,0;161,1;162,0;166,1;181,0;187,1;199,0;211,-1; 3:8,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = text; inactive; preserveAspect] a) Solve the equations! ;[s] 2:0,1;2,0;24,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[{x == t, y == -t, z == -4t, x - 2y + z == 6}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -6, y -> 6, z -> 24, t -> -6}} ;[o] {{x -> -6, y -> 6, z -> 24, t -> -6}} :[font = text; inactive; preserveAspect] The point is {-6,6,24}. ;[s] 3:0,0;13,1;22,0;24,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] b) ;[s] 2:0,1;2,0;4,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; startGroup] Solve[{x - 2y + z == 6, 2x + y - z == 2, y + 3z == 4}] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 7/3, y -> -1, z -> 5/3}} ;[o] 7 5 {{x -> -, y -> -1, z -> -}} 3 3 :[font = text; inactive; preserveAspect; endGroup] the point is {7/3, -1, 5/3}. ;[s] 3:0,0;13,1;27,0;29,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; ^*)