Gerade durch Ebene
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Exercise:
In welchem Punkt durchstösst die Gerade g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix die Ebene mathcalE: Ex x Ey y Ez z Eih?
Solution:
tdplotsetmaincoords center tikzpicturelatex scale. tdplot_main_coords tikzsetscaled unit vectors. foreach x in --... drawcolorgray scaled cs x---x; foreach y in --... drawcolorgray scaled cs -y--y; drawcolorgreen!!black- scaled cs --- noderight small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeleft small bmz; %drawdotted scaled cs PxXPyXPzX--PxXPyX; drawdotted scaled cs DxXDyXDzX--DxXDyX; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX nodeleftsmallg; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX noderight tiny vec vpmatrix tx ty tzpmatrix; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--..--CX--cycle; drawcolorblue thick scaled cs PxXPyXPzX--+tX*txXtX*tyXtX*tzX; drawcolorblue scaled cs dashed PxX+tX*txXPyX+tX*tyXPzX+tX*tzX--+-.*txX-.*tyX-.*tzX; draw- stealth colorred!!yellow thick scaled cs --+PxXPyXPzX nodemidwayabove tiny vec r_P; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!white coordinatesPxXPyXPzX noderightred!!yellow tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX nodeleft tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX noderight tiny D; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX noderight tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinates-AXBXCX noderight tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX noderight tiny Q; tikzpicture center Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x +t y -t z -t cases center +t--t--t - +t--t--t+ +t-+t-+t+ +t t - t tX Dieses t wird nun bei den oberen drei Komponentengleichungen eingesetzt so erhält man schliesslich den Durchstosspunkt Q. center t einsetzen cases x + tX y -tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt Q||.
In welchem Punkt durchstösst die Gerade g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix die Ebene mathcalE: Ex x Ey y Ez z Eih?
Solution:
tdplotsetmaincoords center tikzpicturelatex scale. tdplot_main_coords tikzsetscaled unit vectors. foreach x in --... drawcolorgray scaled cs x---x; foreach y in --... drawcolorgray scaled cs -y--y; drawcolorgreen!!black- scaled cs --- noderight small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeleft small bmz; %drawdotted scaled cs PxXPyXPzX--PxXPyX; drawdotted scaled cs DxXDyXDzX--DxXDyX; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX nodeleftsmallg; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX noderight tiny vec vpmatrix tx ty tzpmatrix; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--..--CX--cycle; drawcolorblue thick scaled cs PxXPyXPzX--+tX*txXtX*tyXtX*tzX; drawcolorblue scaled cs dashed PxX+tX*txXPyX+tX*tyXPzX+tX*tzX--+-.*txX-.*tyX-.*tzX; draw- stealth colorred!!yellow thick scaled cs --+PxXPyXPzX nodemidwayabove tiny vec r_P; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!white coordinatesPxXPyXPzX noderightred!!yellow tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX nodeleft tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX noderight tiny D; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX noderight tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinates-AXBXCX noderight tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX noderight tiny Q; tikzpicture center Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x +t y -t z -t cases center +t--t--t - +t--t--t+ +t-+t-+t+ +t t - t tX Dieses t wird nun bei den oberen drei Komponentengleichungen eingesetzt so erhält man schliesslich den Durchstosspunkt Q. center t einsetzen cases x + tX y -tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt Q||.