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" }{TEXT 225 24 "Work all the exercises. " }}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{SECT 1 {PARA 214 "" 0 "" {TEXT 204 16 "Examples, par t 1" }}{PARA 206 "" 0 "" {TEXT 225 157 "To describe a line in space, w e only need a point P on the line, and a direction in which the line r uns. Of course, we specify that direction with a vector " }{TEXT 229 1 "v" }{TEXT 225 119 ". To get from the origin O to any point Q on th e line, think of first going from O to P and then some distance in the " }{TEXT 229 1 "v" }{TEXT 225 67 "-direction. Thus, a vector equatio n of the line through the point " }}{PARA 206 "" 0 "" {TEXT 225 1 "(" }{XPPEDIT 19 1 "x[0];" "6#&I\"xG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 225 1 " " }{TEXT 225 1 "," }{XPPEDIT 19 1 "y[0];" "6#&I\"yG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 225 1 " " }{TEXT 225 1 "," }{XPPEDIT 19 1 "z[ 0];" "6#&I\"zG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 225 1 " " }{TEXT 225 1 ")" }{TEXT 225 16 " in the directi" }{TEXT 225 18 "on of the ve ctor <" }{TEXT 215 1 "a" }{TEXT 225 2 ", " }{TEXT 215 1 "b" }{TEXT 225 2 ", " }{TEXT 215 1 "c" }{TEXT 225 5 "> is" }}{PARA 206 "" 0 "" {TEXT 225 28 " < " }{TEXT 215 1 "x" }{TEXT 225 2 ", " }{TEXT 215 1 "y" }{TEXT 225 2 ", " }{TEXT 215 1 "z" }{TEXT 225 5 "> = <" }{XPPEDIT 19 1 "x[0];" "6#&I\"xG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " }{TEXT 225 1 "," }{TEXT 215 1 " " }{XPPEDIT 19 1 "y[0];" "6#&I\"yG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " } {TEXT 225 2 ", " }{XPPEDIT 19 1 "z[0];" "6#&I\"zG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " }{TEXT 225 4 "> + " }{TEXT 215 1 "t" }{TEXT 225 2 " <" }{TEXT 215 1 "a" }{TEXT 225 2 ", " }{TEXT 215 1 "b" }{TEXT 225 2 ", " }{TEXT 215 1 "c" }{TEXT 225 5 "> = <" }{XPPEDIT 19 1 "x[0] ;" "6#&I\"xG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " }{TEXT 225 3 " + " }{TEXT 215 1 "a" }{TEXT 225 1 " " }{TEXT 215 1 "t" }{TEXT 225 2 ", " }{XPPEDIT 19 1 "y[0];" "6#&I\"yG6\"6#\"\"!" }{TEXT 254 1 " " } {TEXT 251 1 " " }{TEXT 225 2 "+ " }{TEXT 215 1 "b" }{TEXT 215 2 " t" } {TEXT 225 2 ", " }{XPPEDIT 19 1 "z[0];" "6#&I\"zG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " }{TEXT 225 3 " + " }{TEXT 215 1 "c" }{TEXT 225 1 " " }{TEXT 215 1 "t" }{TEXT 225 1 ">" }}{PARA 212 "" 0 "" {TEXT 219 1 " " }}{PARA 206 "" 0 "" {TEXT 225 21 "The idea is that, as " } {XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 53 " runs through the real numbers, the position vector <" }{TEXT 215 1 "x" }{TEXT 225 2 ", " }{TEXT 215 1 "y" }{TEXT 225 2 ", " }{TEXT 215 1 "z" }{TEXT 225 33 "> traces out the line in space." }}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{PARA 206 "" 0 "" {TEXT 225 23 "To commun icate this to " }{TEXT 215 5 "Maple" }{TEXT 225 69 ", we use square br aces to represent both points and vectors. We use " }{TEXT 244 5 "eva lm" }{TEXT 225 10 " to force " }{TEXT 215 5 "Maple" }{TEXT 225 35 " to perform operations, and we use " }{TEXT 244 2 ":=" }{TEXT 225 20 " to assign the name " }{TEXT 244 4 "line" }{TEXT 225 126 " to the result \+ so we can refer to it later. So the line through (1,2,3) in the direc tion of <2,-1,1> is described as follows:" }}{EXCHG {PARA 206 "> " 0 " " {MPLTEXT 1 280 34 "line := evalm([1,2,3]+t*[2,-1,1]);" }}}{EXCHG {PARA 206 "> " 0 "" {TEXT 282 0 "" }}}{PARA 206 "" 0 "" {TEXT 225 12 " The command " }{TEXT 244 10 "spacecurve" }{TEXT 225 52 " will plot (pa rt of) this line in 3 dimensions, for " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 30 " between any two real numbers:" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 12 "with(plots) :" }}{PARA 206 "> " 0 "" {MPLTEXT 1 280 101 "spacecurve(line, t=0..3, \+ axes=boxed, shading=zhue, thickness=2, labels=[x,y,z], scaling=constra ined);" }}}{EXCHG {PARA 206 "> " 0 "" {TEXT 282 0 "" }}}{PARA 206 "" 0 "" {TEXT 225 125 "By rotating the plot if necessary you can verify th at the line indeed contains the point (1,2,3) (corresponding to the va lue " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 41 ") and runs in the direction of <2,-1,1>. " }} {PARA 206 "" 0 "" {TEXT 225 51 "To describe a plane in space, we only \+ need a point " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 27 " on the plane and a vector " }{TEXT 229 1 "n" }{TEXT 225 57 " to describe which way the plane is facing. The vecto r " }{TEXT 229 1 "n" }{TEXT 273 1 " " }{TEXT 225 12 "is called a " } {TEXT 215 13 "normal vector" }{TEXT 225 64 " for the plane; it is orth ogonal to the plane itself. Thus, if " }{XPPEDIT 18 0 "Q;" "6#%\"QG" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 44 " is any point in the \+ plane, then the vector " }{XPPEDIT 18 0 "PQ;" "6#%#PQG" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 23 " must be orthogonal to " }{TEXT 229 1 "n" }{TEXT 225 21 ". This leads to the " }{TEXT 215 28 "scalar \+ equation of the plane" }{TEXT 225 20 " through the point (" }{XPPEDIT 19 1 "x[0];" "6#&I\"xG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " } {TEXT 225 2 ", " }{XPPEDIT 19 1 "y[0];" "6#&I\"yG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " }{TEXT 225 1 "," }{XPPEDIT 19 1 "z[0];" "6# &I\"zG6\"6#\"\"!" }{TEXT 254 1 " " }{TEXT 251 1 " " }{TEXT 225 2 " )" }{TEXT 225 34 " and orthogonal to the vector <" }{TEXT 215 1 "a" } {TEXT 225 2 ", " }{TEXT 215 1 "b" }{TEXT 225 2 ", " }{TEXT 215 1 "c" } {TEXT 225 4 "> : " }}{PARA 212 "" 0 "" {XPPEDIT 18 0 "a*(x-x[0])+b*(y- y[0])+c*(z-z[0]) = 0;" "6#/,(*&%\"aG\"\"\",&%\"xGF'&F)6#\"\"!!\"\"F'F' *&%\"bGF',&%\"yGF'&F1F+F-F'F'*&%\"cGF',&%\"zGF'&F6F+F-F'F'F," }{TEXT 219 1 " " }{TEXT 219 1 " " }{TEXT 219 1 "." }}{PARA 206 "" 0 "" {TEXT 225 14 "Alternatively," }}{PARA 212 "" 0 "" {XPPEDIT 18 0 "a*x+b*y+c*z = d;" "6#/,(*&%\"aG\"\"\"%\"xGF'F'*&%\"bGF'%\"yGF'F'*&%\"cGF'%\"zGF'F '%\"dG" }{TEXT 219 1 " " }{TEXT 219 1 " " }{TEXT 219 2 ", " }}{PARA 206 "" 0 "" {TEXT 225 6 "where " }{XPPEDIT 18 0 "d = a*x[0]+b*y[0]+c*z [0];" "6#/%\"dG,(*&%\"aG\"\"\"&%\"xG6#\"\"!F(F(*&%\"bGF(&%\"yGF+F(F(*& %\"cGF(&%\"zGF+F(F(" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 3 ". " }}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{PARA 206 "" 0 "" {TEXT 225 16 "Recall that the " }{TEXT 215 5 "Maple" }{TEXT 225 9 " command " } {TEXT 244 14 "implicitplot3d" }{TEXT 225 181 " can be used to plot the set of all points satisfying such an equation. For example, here is \+ (a piece of) the plane through the point (1,2,3) and orthogonal to the vector <2,-1,1>:" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 47 "pla ne:= (2)*(x-1) + (-1)*(y-2) + (1)*(z-3) = 0;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 92 "implicitplot3d(plane, x=0..3, y=0..3, z=0..3, axes=boxed, shading=zhue, style=patchcontour);" }}}{PARA 221 "" 0 "" {TEXT 283 0 "" }}{PARA 206 "" 0 "" {TEXT 225 139 "Again, you should vi sually check that this plane has the desired properties. (using normal axis and finding the intercepts will be helpful)" }}{PARA 206 "" 0 "" {TEXT 225 56 "For some more practice consider the following examples: " }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 56 "implicitplot3d(z=3, x=0..5, y=0..5, z=0..5, axes=boxed);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 61 "implicitplot3d(x+y=1, x=-3..3, y=-3..3, z=-3..3, ax es=boxed);" }}}}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{SECT 1 {PARA 208 " " 0 "" {TEXT 275 55 "Exercises, part 1, to turn in on or before Feb 6 , 2008" }}{EXCHG {PARA 206 "" 0 "" {TEXT 229 2 "1." }{TEXT 225 123 " F ind the vector equation of the line through (1,0,-1) with direction ve ctor <1,2,0>. Write the curve in Maple and plot it." }}}{EXCHG {PARA 206 "" 0 "" {TEXT 229 2 "2." }{TEXT 225 104 " Find the equation of the plane containing the points A(1,-3,2), B(0,4,2) and C(-2,1,1). Plot t he plane." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 0 "" }}{PARA 206 "" 0 "" {TEXT 282 13 "Solution: 1. " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 246 11 " Solution: 2" }}}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 208 "" 0 "" {TEXT 275 16 "Examples, part 2" }}{PARA 206 "" 0 "" {TEXT 225 30 "The only \+ command we'll use is " }{TEXT 244 14 "implicitplot3d" }{TEXT 225 171 " . Recall that this command plots the set of all points which satisfy \+ a given equation in three variables. For instance, the paraboloid of \+ points satisfying the equation " }{XPPEDIT 18 0 "y = x^2+z^2;" "6#/%\" yG,&*$)%\"xG\"\"#\"\"\"F**$)%\"zGF)F*F*" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 23 " is plotted as follows:" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 123 "implicitplot3d(y = x^2 + z^2, x=-2..2, y=0..4 , z=-2..2, style=patchcontour, shading=zhue, axes=frame, scaling=const rained);" }}}{EXCHG {PARA 206 "> " 0 "" {TEXT 282 0 "" }}}{PARA 206 "" 0 "" {TEXT 225 35 "Notice that I've chosen the option " }{TEXT 244 18 "style=patchcontour" }{TEXT 225 61 " to get contour lines drawn on \+ the surface. Also, I've used " }{TEXT 244 12 "shading=zhue" }{TEXT 225 255 " so that the color of the surface corresponds to its height. \+ You can change the options in any plot display by right-clicking on the plot and selecting from the menu that appears, and you can rotate it by left-grabbing the plot and moving the mouse. " }}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{PARA 206 "" 0 "" {TEXT 225 91 "In particular, \+ by rotating the plot until it is viewed from above - or by using the o ption " }{TEXT 244 19 "orientation=[-90,0]" }{TEXT 225 97 " - you effe ctively get a contour diagram for the surface. (Careful, it can be tr icky to get the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 225 1 " " } {TEXT 225 1 " " }{TEXT 225 6 "- and " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 100 "-axes running the way you're used to.) In the example above, you should see the contour pa rabolas " }{XPPEDIT 18 0 "y = x^2+k^2;" "6#/%\"yG,&*$)%\"xG\"\"#\"\" \"F**$)%\"kGF)F*F*" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 39 " \+ corresponding to different choices of " }{XPPEDIT 18 0 "z = k;" "6#/% \"zG%\"kG" }{TEXT 225 1 " " }{TEXT 225 1 " " }{TEXT 225 3 ". " }} {PARA 206 "" 0 "" {TEXT 225 37 "Just to warm up see one more example:" }}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 280 69 "implicitplot3d(x^2+y^2+z^2=1, x=-1..1, y=-1..1, z=- 1..1, axes=boxed);" }}}{EXCHG {PARA 205 "> " 0 "" {TEXT 285 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {TEXT 282 0 "" }}}{EXCHG {PARA 206 "> " 0 " " {TEXT 282 0 "" }}}}{SECT 1 {PARA 208 "" 0 "" {TEXT 275 95 "Exercises , part 2(you should choose numerical values for a, b, c and do any thr ee of the given)" }{TEXT 275 37 ", to turn in on or before Feb 6, 2008 " }}{PARA 206 "" 0 "" {TEXT 225 5 "Plot:" }}{PARA 206 "" 0 "" {TEXT 282 0 "" }}{PARA 212 "" 0 "" {TEXT 211 3 "A. " }{TEXT 219 2 " " } {XPPEDIT 18 0 "x^2/(a^2)+y^2/(b^2)+z^2/(c^2) = 1;" "6#/,(*&)%\"xG\"\"# \"\"\"*$)%\"aGF(F)!\"\"F)*&)%\"yGF(F)*$)%\"bGF(F)F-F)*&)%\"zGF(F)*$)% \"cGF(F)F-F)F)" }{TEXT 219 1 " " }{TEXT 219 1 " " }{TEXT 211 14 " \+ B. " }{TEXT 219 2 " " }{XPPEDIT 18 0 "x^2/(a^2)+y^2/(b^2)-z^2/( c^2) = 1;" "6#/,(*&)%\"xG\"\"#\"\"\"*$)%\"aGF(F)!\"\"F)*&)%\"yGF(F)*$) %\"bGF(F)F-F)*&)%\"zGF(F)*$)%\"cGF(F)F-F-F)" }{TEXT 219 1 " " }{TEXT 219 1 " " }{TEXT 211 3 " " }}{PARA 212 "" 0 "" {TEXT 211 2 "C." } {TEXT 219 4 " " }{XPPEDIT 18 0 "x^2/(a^2)-y^2/(b^2)-z^2/(c^2) = 1;" "6#/,(*&)%\"xG\"\"#\"\"\"*$)%\"aGF(F)!\"\"F)*&)%\"yGF(F)*$)%\"bGF(F)F -F-*&)%\"zGF(F)*$)%\"cGF(F)F-F-F)" }{TEXT 219 1 " " }{TEXT 219 1 " " } {TEXT 211 12 " D." }{TEXT 219 4 " " }{XPPEDIT 18 0 "x^2/(a ^2)+y^2/(b^2) = z^2/(c^2);" "6#/,&*&)%\"xG\"\"#\"\"\"*$)%\"aGF(F)!\"\" F)*&)%\"yGF(F)*$)%\"bGF(F)F-F)*&)%\"zGF(F)*$)%\"cGF(F)F-" }{TEXT 219 1 " " }{TEXT 219 1 " " }{TEXT 211 3 " " }}{PARA 212 "" 0 "" {TEXT 211 2 "E." }{TEXT 219 3 " " }{XPPEDIT 18 0 "x^2/(a^2)+y^2/(b^2) = z/ c;" "6#/,&*&)%\"xG\"\"#\"\"\"*$)%\"aGF(F)!\"\"F)*&)%\"yGF(F)*$)%\"bGF( F)F-F)*&%\"zGF)%\"cGF-" }{TEXT 219 1 " " }{TEXT 219 1 " " }{TEXT 211 14 " F. " }{TEXT 219 3 " " }{XPPEDIT 18 0 "x^2/(a^2)-y^2/( b^2) = z/c;" "6#/,&*&)%\"xG\"\"#\"\"\"*$)%\"aGF(F)!\"\"F)*&)%\"yGF(F)* $)%\"bGF(F)F-F-*&%\"zGF)%\"cGF-" }{TEXT 219 1 " " }{TEXT 219 1 " " } {TEXT 219 2 " " }}{PARA 212 "" 0 "" {TEXT 219 0 "" }}{PARA 0 "" 0 "" {TEXT 219 10 "Solution: " }}{PARA 206 "" 0 "" {TEXT 282 15 "Example fo r A. " }{TEXT 219 2 " " }{XPPEDIT 2 0 "x^2/(a^2)+y^2/(b^2)+z^2/(c^2) \+ = 1;" "6#/,(*&)%\"xG\"\"#\"\"\"*$)%\"aGF(F)!\"\"F)*&)%\"yGF(F)*$)%\"bG F(F)F-F)*&)%\"zGF(F)*$)%\"cGF(F)F-F)F)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a := 2: b := 3: c := 4:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "implicitplot3d(x^2/a^2+y^ 2/b^2 + z^2/c^2=1, x=-5..5, y=-5..5, z=-5..5, style=patchcontour, shad ing=zhue, axes=normal, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "" "%#%?G" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }