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; drawcolorblue dashed scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue dashed scaled cs PxXPyXPzX--+*txX*tyX*tzX; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX---.---.--cycle; drawcolorblue scaled cs ---+-*txX-*tyX-*tzX; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwayright tiny vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX noderight tiny PPx|Py|Pz; drawdotted scaled cs ----; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinates- nodeleft tiny D; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX nodebelow tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodebelow tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX noderight tiny C; tikzpicture center 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; drawcolorblue dashed scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue dashed scaled cs PxXPyXPzX--+*txX*tyX*tzX; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX---.---.--cycle; drawcolorblue scaled cs ---+-*txX-*tyX-*tzX; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwayright tiny vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX noderight tiny P; drawdotted scaled cs ----; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinates- nodeleft tiny D; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX nodebelow tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX noderight tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX node right tiny C; 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 D. center t einsetzen cases x + tX y -+ tX z - tX cases center Das führt dann schliesslich auf den Durchstosspunkt D|-|.