(*^ ::[ 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-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 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; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; fontName = "CalcMath"; startGroup] The Intersection Of Three Planes :[font = subsubtitle; inactive; preserveAspect] By: Debra Woods 12/1993 :[font = section; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] Introduction to three dimensional space :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] You are already familiar with the 2-dimensional coordinate system, that is ordered pairs. We will now add another dimension, and investigate the 3-dimensional coordinate system. The 3-D coordinate system takes the xy-plane that you are familiar with and adds another axis that is orthogonal (perpendicular) the both the x-axis and the y-axis. In this lesson, we will investigate the graphs of points, planes, and lines in space. To start, let's look at the 3-D axes. :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] h=5; spacer=h/10; threedims= Graphics3D[{ {Line[{{-h,0,0},{h,0,0}}]}, Text["x",{h + spacer,0,0}], {Line[{{0,-h,0},{0,h,0}}]}, Text["y",{0, h+spacer,0}], {Line[{{0,0,-h},{0,0,h}}]}, Text["z",{0,0,h + spacer}]}, Boxed->False,Axes->False, ViewPoint->{1.75,1.224,1.7} ,AspectRatio->Automatic]; Show[threedims]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 272; pictureHeight = 287; fontName = "CalcMath"; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.05251 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0229969 1.08368 0.027092 1.08368 [ [ 0 0 0 0 ] [ 1 1.05251 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.05251 L 0 1.05251 L closepath clip newpath p P p .004 w .65518 .69403 m .34312 .41703 L s P p .004 w .51584 .37926 m .51516 .80424 L s P p .004 w .3107 .65932 m .75285 .46667 L s P p P p [(x)] .32363 .39972 0 0 Mshowa [(y)] .77866 .45543 0 0 Mshowa [(z)] .51512 .83062 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; fontName = "CalcMath"] The points in 3-D space are ordered triples (x, y, z). The following example shows plots of the points (2,3,1) in red, and (3,-2,-1) in green. :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] pts = Graphics3D[{{RGBColor[1,0,0],PointSize[0.02], Point[{2,3,1}]},{RGBColor[0,1,0], PointSize[0.02],Point[{3,-2,-1}]}}]; Show[threedims,pts]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 272; pictureHeight = 287; fontName = "CalcMath"; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.05251 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0229969 1.08368 0.027092 1.08368 [ [ 0 0 0 0 ] [ 1 1.05251 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.05251 L 0 1.05251 L closepath clip newpath p P p .004 w .65518 .69403 m .34312 .41703 L s P p .004 w .51584 .37926 m .51516 .80424 L s P p .004 w .3107 .65932 m .75285 .46667 L s P p 0 1 0 r .02 w .33265 .483 Mdot P p 1 0 0 r .02 w .59403 .49149 Mdot P p P p [(x)] .32363 .39972 0 0 Mshowa [(y)] .77866 .45543 0 0 Mshowa [(z)] .51512 .83062 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"; endGroup] If you would like to plot some points of your own, change the coordinates of the points in the above mathematica code. Try changing each coordinate separately to see what happens. :[font = section; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] Introduction to planes in space :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] Planes in space have equations of the form Ax + By + Cz = D. Let's look at a plane. In particular, the plane with equation 6x + 4y + 3z = 12. :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] planeplot = Graphics3D[Polygon[{{2,0,0},{0,3,0},{0,0,4}}], DisplayFunction->Identity]; Show[threedims,planeplot,DisplayFunction->$DisplayFunction, Boxed->False,Axes->True]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 272; pictureHeight = 287; fontName = "CalcMath"; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.05251 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0229969 1.08368 0.027092 1.08368 [ [(-5)] .44233 1.03456 .03029 -1 Msboxa [(-2.5)] .36189 .99028 .06325 -1 Msboxa [(0)] .27263 .94113 .10123 -1 Msboxa [(2.5)] .17302 .88626 .14547 -1 Msboxa [(5)] .06115 .82462 .19766 -1 Msboxa [(-5)] .13028 .2939 1 .85218 Msboxa [(-2.5)] .22467 .23704 1 .93019 Msboxa [(0)] .32613 .17595 .97664 1 Msboxa [(2.5)] .43547 .11015 .87823 1 Msboxa [(5)] .55365 .03908 .77982 1 Msboxa [(-5)] .11876 .30878 1 .83894 Msboxa [(-2.5)] .09873 .40074 1 .78128 Msboxa [(0)] .07656 .50253 1 .71886 Msboxa [(2.5)] .0519 .61583 1 .65107 Msboxa [(5)] .02432 .7427 1 .57719 Msboxa [ 0 0 0 0 ] [ 1 1.05251 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .45069 1.02542 m .02709 .79103 L s P p .002 w .44274 1.02102 m .44295 1.01425 L s P [(-5)] .44233 1.03456 .03029 -1 Mshowa p .002 w .36275 .97676 m .36317 .97 L s P [(-2.5)] .36189 .99028 .06325 -1 Mshowa p .002 w .274 .92765 m .27468 .92091 L s P [(0)] .27263 .94113 .10123 -1 Mshowa p .002 w .17497 .87286 m .17595 .86616 L s P [(2.5)] .17302 .88626 .14547 -1 Mshowa p .002 w .06378 .81133 m .06509 .80469 L s P [(5)] .06115 .82462 .19766 -1 Mshowa p .001 w .42738 1.01252 m .42753 1.00846 L s P p .001 w .41171 1.00385 m .41189 .99979 L s P p .001 w .39572 .995 m .39592 .99095 L s P p .001 w .3794 .98598 m .37963 .98192 L s P p .001 w .34574 .96735 m .34603 .9633 L s P p .001 w .32838 .95774 m .32869 .95369 L s P p .001 w .31064 .94793 m .31098 .94388 L s P p .001 w .29252 .9379 m .29289 .93385 L s P p .001 w .25507 .91718 m .25551 .91314 L s P p .001 w .23572 .90647 m .23619 .90243 L s P p .001 w .21593 .89552 m .21644 .89149 L s P p .001 w .19568 .88432 m .19623 .88029 L s P p .001 w .15377 .86113 m .1544 .85711 L s P p .001 w .13207 .84912 m .13274 .84511 L s P p .001 w .10985 .83683 m .11056 .83283 L s P p .001 w .0871 .82424 m .08784 .82024 L s P p .001 w .03988 .79811 m .04071 .79413 L s P P p p .002 w .1311 .30838 m .59974 .02709 L s P p .002 w .14059 .30268 m .14575 .30708 L s P [(-5)] .13028 .2939 1 .85218 Mshowa p .002 w .23459 .24626 m .23955 .25088 L s P [(-2.5)] .22467 .23704 1 .93019 Mshowa p .002 w .33559 .18564 m .34033 .19049 L s P [(0)] .32613 .17595 .97664 1 Mshowa p .002 w .44441 .12033 m .44888 .12542 L s P [(2.5)] .43547 .11015 .87823 1 Mshowa p .002 w .56198 .04976 m .56614 .0551 L s P [(5)] .55365 .03908 .77982 1 Mshowa p .001 w .15887 .29171 m .16194 .29438 L s P p .001 w .1774 .28059 m .18045 .28328 L s P p .001 w .19619 .26931 m .19922 .27202 L s P p .001 w .21526 .25787 m .21826 .26061 L s P p .001 w .25421 .23449 m .25716 .23728 L s P p .001 w .27411 .22254 m .27703 .22537 L s P p .001 w .2943 .21042 m .2972 .21327 L s P p .001 w .3148 .19812 m .31766 .201 L s P p .001 w .3567 .17297 m .35951 .17591 L s P p .001 w .37813 .16011 m .38091 .16307 L s P p .001 w .39989 .14705 m .40263 .15004 L s P p .001 w .42198 .13379 m .42469 .13681 L s P p .001 w .46719 .10665 m .46983 .10974 L s P p .001 w .49033 .09277 m .49294 .09588 L s P p .001 w .51383 .07866 m .51641 .0818 L s P p .001 w .53771 .06432 m .54025 .0675 L s P p .001 w .58663 .03496 m .58909 .03819 L s P P p p .002 w .1311 .30838 m .02709 .79103 L s P p .002 w .12914 .31749 m .13433 .32184 L s P [(-5)] .11876 .30878 1 .83894 Mshowa p .002 w .1094 .40908 m .11474 .41325 L s P [(-2.5)] .09873 .40074 1 .78128 Mshowa p .002 w .08756 .51044 m .09306 .51439 L s P [(0)] .07656 .50253 1 .71886 Mshowa p .002 w .06325 .62322 m .06893 .62691 L s P [(2.5)] .0519 .61583 1 .65107 Mshowa p .002 w .03605 .74947 m .04191 .75286 L s P [(5)] .02432 .7427 1 .57719 Mshowa p .001 w .12534 .33509 m .12847 .33768 L s P p .001 w .12148 .35304 m .12462 .35561 L s P p .001 w .11753 .37135 m .1207 .3739 L s P p .001 w .11351 .39002 m .11669 .39255 L s P p .001 w .10521 .42852 m .10843 .431 L s P p .001 w .10093 .44836 m .10417 .45082 L s P p .001 w .09657 .46862 m .09983 .47105 L s P p .001 w .09211 .48931 m .09539 .49171 L s P p .001 w .08291 .53202 m .08623 .53436 L s P p .001 w .07815 .55407 m .0815 .55639 L s P p .001 w .0733 .57661 m .07666 .5789 L s P p .001 w .06833 .59966 m .07172 .60191 L s P p .001 w .05806 .64732 m .06149 .6495 L s P p .001 w .05275 .67197 m .0562 .67412 L s P p .001 w .04731 .6972 m .05078 .69931 L s P p .001 w .04175 .72303 m .04524 .7251 L s P p .001 w .03021 .77655 m .03376 .77854 L s P P 0 0 m 1 0 L 1 1.05251 L 0 1.05251 L closepath clip newpath p P p .002 w .45111 .51288 m .65354 .50994 L .51524 .75325 L closepath p .697 .507 .626 r F P s P p .004 w .65354 .50994 m .75285 .46667 L s P p .004 w .3107 .65932 m .47149 .58926 L s P p .004 w .51516 .80424 m .51524 .75325 L s P p .004 w .51563 .51194 m .51584 .37926 L s P p .004 w .34312 .41703 m .45111 .51288 L s P p .004 w .58454 .63133 m .65518 .69403 L s P p P p p .002 w .1311 .30838 m .02709 .79103 L s P p .002 w .12914 .31749 m .13433 .32184 L s P [(-5)] .11876 .30878 1 .83894 Mshowa p .002 w .1094 .40908 m .11474 .41325 L s P [(-2.5)] .09873 .40074 1 .78128 Mshowa p .002 w .08756 .51044 m .09306 .51439 L s P [(0)] .07656 .50253 1 .71886 Mshowa p .002 w .06325 .62322 m .06893 .62691 L s P [(2.5)] .0519 .61583 1 .65107 Mshowa p .002 w .03605 .74947 m .04191 .75286 L s P [(5)] .02432 .7427 1 .57719 Mshowa p .001 w .12534 .33509 m .12847 .33768 L s P p .001 w .12148 .35304 m .12462 .35561 L s P p .001 w .11753 .37135 m .1207 .3739 L s P p .001 w .11351 .39002 m .11669 .39255 L s P p .001 w .10521 .42852 m .10843 .431 L s P p .001 w .10093 .44836 m .10417 .45082 L s P p .001 w .09657 .46862 m .09983 .47105 L s P p .001 w .09211 .48931 m .09539 .49171 L s P p .001 w .08291 .53202 m .08623 .53436 L s P p .001 w .07815 .55407 m .0815 .55639 L s P p .001 w .0733 .57661 m .07666 .5789 L s P p .001 w .06833 .59966 m .07172 .60191 L s P p .001 w .05806 .64732 m .06149 .6495 L s P p .001 w .05275 .67197 m .0562 .67412 L s P p .001 w .04731 .6972 m .05078 .69931 L s P p .001 w .04175 .72303 m .04524 .7251 L s P p .001 w .03021 .77655 m .03376 .77854 L s P P p [(x)] .32363 .39972 0 0 Mshowa [(y)] .77866 .45543 0 0 Mshowa [(z)] .51512 .83062 0 0 Mshowa P % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] The intersection of planes :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] Intersection of two planes :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] Of particular interest is the intersection of planes. First, let's investigate the intersection of two planes. :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] plane1 = Graphics3D[Polygon[{{0,2,2},{0,2,-2},{0,-2,-2}, {0,-2,2}}]]; plane2 = Graphics3D[Polygon[{{2,2,0},{2,-2,0},{-2,-2,0}, {-2,2,0}}]]; plane3 = Graphics3D[Polygon[{{3,2,2},{3,2,-2},{3,-2,-2}, {3,-2,2}}]]; Do[Show[GraphicsArray[{ Show[plane1, plane2,Boxed->False,Axes->False, DisplayFunction->Identity], Show[plane1, plane3,Boxed->False,Axes->False, DisplayFunction->Identity]}], DisplayFunction->$DisplayFunction]]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 134; fontName = "CalcMath"; endGroup] %! %%Creator: Mathematica %%AspectRatio: .47619 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0113379 0.47619 [ [ 0 0 0 0 ] [ 1 .47619 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .47619 L 0 .47619 L closepath clip newpath p p % Start of sub-graphic p 0.0238095 0.0113379 0.477324 0.464853 MathSubStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0260524 1.10184 0.0275459 1.10184 [ [ 0 0 0 0 ] [ 1 1.08079 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 0 m 1 0 L 1 1.08079 L 0 1.08079 L closepath clip newpath p P p .002 w .09571 .50305 m .41935 .78823 L .65569 .72122 L p .656 .751 .92 r F P s P p .002 w .65569 .72122 m .64649 .50519 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .65569 .72122 m .91647 .64728 L p .52487 .57846 L .656 .751 .92 r F P s P p .52487 .57846 m .09571 .50305 L .65569 .72122 L .656 .751 .92 r F P p .002 w .64649 .50519 m .37855 .171 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .6664 .9726 m .65569 .72122 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .34875 .68456 m .6664 .9726 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .37855 .171 m .36503 .40405 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .36503 .40405 m .09571 .50305 L p .52487 .57846 L .656 .751 .92 r F P s P p .002 w .36503 .40405 m .34875 .68456 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .91647 .64728 m .66876 .2924 L .36503 .40405 L p .656 .751 .92 r F P s P p .91647 .64728 m .36503 .40405 L .52487 .57846 L .656 .751 .92 r F P p P p P MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.522676 0.0113379 0.97619 0.464853 MathSubStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0819831 1.22796 0.030699 1.22796 [ [ 0 0 0 0 ] [ 1 1.2198 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 0 m 1 0 L 1 1.2198 L 0 1.2198 L closepath clip newpath p P p .002 w .47005 1.16458 m .47847 .62254 L .13546 .25531 L p .945 .686 .513 r F P s P p .002 w .94797 1.05313 m .8874 .48883 L .60146 .06171 L p .976 .674 .354 r F P s P p .002 w .13546 .25531 m .05854 .8523 L .47005 1.16458 L p .945 .686 .513 r F P s P p .002 w .60146 .06171 m .61634 .67957 L .94797 1.05313 L p .976 .674 .354 r F P s P p P p P MathSubEnd P % End of sub-graphic P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"; endGroup] Notice that the intersection of two planes is either a line, if the planes are not parallel, or the empty set if the planes are parallel. :[font = subsection; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] Intersection of three planes :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] What would the intersection of three planes be? Let's investigate several cases. :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] plane4 = Graphics3D[Polygon[{{2,0,2},{2,0,-2},{-2,0,-2}, {-2,0,2}}]]; Show[plane1, plane2, plane4,Boxed->False]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 265; pictureHeight = 286; fontName = "CalcMath"; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.08079 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0260524 1.10184 0.0275459 1.10184 [ [ 0 0 0 0 ] [ 1 1.08079 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.08079 L 0 1.08079 L closepath clip newpath p P p .002 w .41935 .78823 m .65569 .72122 L p .39554 .61987 L .656 .751 .92 r F P s P p .002 w .27286 .65915 m .41935 .78823 L p .39554 .61987 L .656 .751 .92 r F P s P p .002 w .65569 .72122 m .64649 .50519 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .65569 .72122 m .91647 .64728 L p .52487 .57846 L .656 .751 .92 r F P s P p .002 w .29129 .43897 m .27286 .65915 L p .52487 .57846 L .544 .324 .532 r F P s P p .52487 .57846 m .09571 .50305 L .65569 .72122 L .656 .751 .92 r F P p .002 w .6664 .9726 m .65569 .72122 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .25127 .91705 m .52487 .84426 L p .52487 .57846 L .544 .324 .532 r F P s P p .002 w .27286 .65915 m .25127 .91705 L p .52487 .57846 L .544 .324 .532 r F P s P p .002 w .09571 .50305 m .27286 .65915 L p .39554 .61987 L .656 .751 .92 r F P s P p .002 w .64649 .50519 m .52487 .35349 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .78299 .25903 m .52487 .35349 L p .66137 .53476 L .544 .324 .532 r F P s P p .002 w .52487 .35349 m .29129 .43897 L p .52487 .57846 L .544 .324 .532 r F P s P p .002 w .91647 .64728 m .80568 .48856 L p .52487 .57846 L .656 .751 .92 r F P s P p .83276 .76234 m .52487 .35349 L .52487 .57846 L .544 .324 .532 r F P p .002 w .52487 .35349 m .37855 .171 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .36503 .40405 m .09571 .50305 L p .52487 .57846 L .656 .751 .92 r F P s P p .002 w .34875 .68456 m .52487 .84426 L p .44914 .49584 L .966 .687 .452 r F P s P p .002 w .37855 .171 m .36503 .40405 L p .44914 .49584 L .966 .687 .452 r F P s P p .002 w .36503 .40405 m .34875 .68456 L p .44914 .49584 L .966 .687 .452 r F P s P p .52487 .57846 m .37855 .171 L .52487 .84426 L .966 .687 .452 r F P p .002 w .66876 .2924 m .36503 .40405 L p .52487 .57846 L .656 .751 .92 r F P s P p .002 w .80568 .48856 m .78299 .25903 L p .66137 .53476 L .544 .324 .532 r F P s P p .002 w .80568 .48856 m .66876 .2924 L p .52487 .57846 L .656 .751 .92 r F P s P p .002 w .83276 .76234 m .80568 .48856 L p .66137 .53476 L .544 .324 .532 r F P s P p .002 w .52487 .84426 m .6664 .9726 L p .52487 .57846 L .966 .687 .452 r F P s P p .002 w .52487 .84426 m .83276 .76234 L p .52487 .57846 L .544 .324 .532 r F P s P p P p P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] The preceeding example shows three planes intersecting at a point. Let's see if there are other possibilities. Could three planes intersect along a line? :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] r = Sqrt[2]; plane5 = Graphics3D[Polygon[{{2,2,2},{2,-2,-2},{-2,-2,-2}, {-2,2,2}}]]; Show[plane5,plane2,plane4,Boxed->False, ViewPoint->{1.200, 1.370, 1.070}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 273; pictureHeight = 276; fontName = "CalcMath"; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.01103 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0681246 1.11581 0.0278953 1.11581 [ [ 0 0 0 0 ] [ 1 1.01103 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.01103 L 0 1.01103 L closepath clip newpath p P p .002 w .50862 .55605 m .67669 .66873 L p .48978 .5858 L .471 0 0 r F P s P p .002 w .14048 .38408 m .50862 .55605 L p .48978 .5858 L .471 0 0 r F P s P p .002 w .1013 .59882 m .51031 .74303 L p .48978 .5858 L .551 .812 .994 r F P s P p .002 w .24797 .47851 m .1013 .59882 L p .48978 .5858 L .551 .812 .994 r F P s P p .002 w .51031 .74303 m .67669 .66873 L p .48978 .5858 L .551 .812 .994 r F P s P p .002 w .65951 .46657 m .69774 .91643 L .21152 .78127 L p .915 .607 .446 r F P s P p .002 w .24797 .47851 m .14048 .38408 L p .48978 .5858 L .471 0 0 r F P s P p .002 w .21152 .78127 m .27597 .24587 L .65951 .46657 L p .915 .607 .446 r F P s P p .002 w .447 .65336 m .24797 .47851 L p .48978 .5858 L .471 0 0 r F P s P p .002 w .45373 .30972 m .24797 .47851 L p .48978 .5858 L .551 .812 .994 r F P s P p .002 w .89261 .5723 m .45373 .30972 L p .48978 .5858 L .551 .812 .994 r F P s P p .002 w .67669 .66873 m .89261 .5723 L p .48978 .5858 L .551 .812 .994 r F P s P p .002 w .94563 .84905 m .447 .65336 L p .48978 .5858 L .471 0 0 r F P s P p .002 w .67669 .66873 m .94563 .84905 L p .48978 .5858 L .471 0 0 r F P s P p P p P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] Yes!! Now, what are the possibilities of three planes not intersecting at all? The obvious answer is if they are all parallel, but could there be other possibilities? Let's see. :[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] plane6 = Graphics3D[Polygon[{{2,1,2},{2,1,-2},{-2,1,-2}, {-2,1,2}}]]; Do[Show[GraphicsArray[{ Show[plane2,plane5,plane6,Boxed->False, ViewPoint->{-3.590, -1.400, 1.070}, DisplayFunction->Identity], Show[plane2,plane6,plane4,Boxed->False, ViewPoint->{-3.590, -1.400, 1.070}, DisplayFunction->Identity]}], DisplayFunction->$DisplayFunction]]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 409; pictureHeight = 194; fontName = "CalcMath"; endGroup] %! %%Creator: Mathematica %%AspectRatio: .47619 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0113379 0.47619 [ [ 0 0 0 0 ] [ 1 .47619 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .47619 L 0 .47619 L closepath clip newpath p p % Start of sub-graphic p 0.0238095 0.0113379 0.477324 0.464853 MathSubStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations 0.00830288 1.05051 0.0262628 1.05051 [ [ 0 0 0 0 ] [ 1 .98835 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 0 m 1 0 L 1 .98835 L 0 .98835 L closepath clip newpath p P p .002 w .06939 .49701 m .36776 .65259 L p .44493 .60555 L .525 .815 .999 r F P s P p .002 w .50394 .35918 m .23982 .13909 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .5031 .64002 m .50394 .35918 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .22959 .47795 m .06939 .49701 L p .44493 .60555 L .525 .815 .999 r F P s P p .002 w .64354 .62697 m .92907 .30368 L p .53356 .55152 L .866 .887 .849 r F P s P p .002 w .94089 .59935 m .75722 .41517 L p .53356 .55152 L .525 .815 .999 r F P s P p .002 w .36776 .65259 m .5031 .64002 L p .44493 .60555 L .525 .815 .999 r F P s P p .44493 .60555 m .64354 .62697 L .53356 .55152 L .525 .815 .999 r F P p .002 w .5031 .64002 m .64354 .62697 L p .44493 .60555 L .525 .815 .999 r F P s P p .44359 .75888 m .5031 .64002 L .38066 .56747 L .963 .757 .577 r F P p .002 w .50267 .78648 m .5031 .64002 L p .44359 .75888 L .963 .757 .577 r F P s P p .44359 .75888 m .64354 .62697 L .53356 .55152 L .866 .887 .849 r F P p .002 w .36303 .94458 m .50267 .78648 L p .44359 .75888 L .866 .887 .849 r F P s P p .002 w .50267 .78648 m .64354 .62697 L p .44359 .75888 L .866 .887 .849 r F P s P p .002 w .05302 .85264 m .36303 .94458 L p .44359 .75888 L .866 .887 .849 r F P s P p .002 w .21861 .84129 m .50222 .93719 L p .44359 .75888 L .963 .757 .577 r F P s P p .44359 .75888 m .38066 .56747 L .30554 .6944 L .963 .757 .577 r F P p .44359 .75888 m .30554 .6944 L .05302 .85264 L .866 .887 .849 r F P p .44359 .75888 m .30554 .6944 L .21861 .84129 L .963 .757 .577 r F P p .002 w .23982 .13909 m .22959 .47795 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .22959 .47795 m .22419 .6564 L p .38066 .56747 L .963 .757 .577 r F P s P p .38066 .56747 m .22419 .6564 L .30554 .6944 L .963 .757 .577 r F P p .44493 .60555 m .53356 .55152 L .22959 .47795 L .525 .815 .999 r F P p .002 w .39725 .458 m .22959 .47795 L p .53356 .55152 L .525 .815 .999 r F P s P p .002 w .39725 .458 m .22419 .6564 L p .53356 .55152 L .866 .887 .849 r F P s P p .002 w .22419 .6564 m .05302 .85264 L p .30554 .6944 L .866 .887 .849 r F P s P p .53356 .55152 m .22419 .6564 L .30554 .6944 L .866 .887 .849 r F P p .002 w .22419 .6564 m .21861 .84129 L p .30554 .6944 L .963 .757 .577 r F P s P p .002 w .64354 .62697 m .94089 .59935 L p .53356 .55152 L .525 .815 .999 r F P s P p .002 w .92907 .30368 m .74915 .05458 L p .53356 .55152 L .866 .887 .849 r F P s P p .002 w .50222 .93719 m .50267 .78648 L p .44359 .75888 L .963 .757 .577 r F P s P p .44359 .75888 m .53356 .55152 L .30554 .6944 L .866 .887 .849 r F P p .002 w .74915 .05458 m .39725 .458 L p .53356 .55152 L .866 .887 .849 r F P s P p .002 w .75722 .41517 m .39725 .458 L p .53356 .55152 L .525 .815 .999 r F P s P p P p P MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.522676 0.0113379 0.97619 0.464853 MathSubStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations 0.00830288 1.05051 0.0262628 1.05051 [ [ 0 0 0 0 ] [ 1 .98835 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 0 m 1 0 L 1 .98835 L 0 .98835 L closepath clip newpath p P p .002 w .06939 .49701 m .36776 .65259 L p .44493 .60555 L .525 .815 .999 r F P s P p .002 w .50394 .35918 m .23982 .13909 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .5031 .64002 m .50394 .35918 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .64046 .34136 m .40195 .11219 L p .53356 .55152 L .979 .766 .547 r F P s P p .002 w .64354 .62697 m .64046 .34136 L p .53356 .55152 L .979 .766 .547 r F P s P p .44493 .60555 m .60932 .6035 L .53356 .55152 L .525 .815 .999 r F P p .002 w .36776 .65259 m .5031 .64002 L p .44493 .60555 L .525 .815 .999 r F P s P p .002 w .5031 .64002 m .94089 .59935 L p .44493 .60555 L .525 .815 .999 r F P s P p .002 w .22959 .47795 m .06939 .49701 L p .44493 .60555 L .525 .815 .999 r F P s P p .002 w .21861 .84129 m .50222 .93719 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .50222 .93719 m .5031 .64002 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .39221 .8294 m .6468 .92952 L p .53356 .55152 L .979 .766 .547 r F P s P p .002 w .40195 .11219 m .39725 .458 L p .53356 .55152 L .979 .766 .547 r F P s P p .002 w .23982 .13909 m .22959 .47795 L p .38066 .56747 L .963 .757 .577 r F P s P p .44493 .60555 m .53356 .55152 L .22959 .47795 L .525 .815 .999 r F P p .002 w .39725 .458 m .22959 .47795 L p .53356 .55152 L .525 .815 .999 r F P s P p .002 w .39725 .458 m .39221 .8294 L p .53356 .55152 L .979 .766 .547 r F P s P p .002 w .94089 .59935 m .75722 .41517 L p .53356 .55152 L .525 .815 .999 r F P s P p .60932 .6035 m .94089 .59935 L .53356 .55152 L .525 .815 .999 r F P p .002 w .22959 .47795 m .21861 .84129 L p .38066 .56747 L .963 .757 .577 r F P s P p .002 w .6468 .92952 m .64354 .62697 L p .53356 .55152 L .979 .766 .547 r F P s P p .002 w .75722 .41517 m .39725 .458 L p .53356 .55152 L .525 .815 .999 r F P s P p P p P MathSubEnd P % End of sub-graphic P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"; endGroup; endGroup; endGroup] To summarize, the intersection of three planes can be a point (that is an ordered triple (a,b,c)), or a line (that is at an infinite number of points), or the null set (no point of intersection). Recall, that the equation of a plane was Ax + By + Cz = D. That is an equation involving three unknown variables. We will now move on the the algebra of solving three equations in three unknowns. ^*)