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https://texercises.com/exercise/gerade-durch-ebene-1/
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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 ----; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--CX--cycle; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwaybelow tiny vec vpmatrix tx ty tzpmatrix; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodebelow tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX noderight tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodeleft tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX nodeabove right tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX nodeleft tiny D; 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 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||.
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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 ----; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX; filldrawcolorblack fillyellow!!white opacity. scaled cs AX--BX--CX--cycle; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidwaybelow tiny vec vpmatrix tx ty tzpmatrix; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodebelow tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesAX noderight tiny A; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesBX nodeleft tiny B; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesCX nodeabove right tiny C; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesDxXDyXDzX nodeleft tiny D; 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 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||.
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Attributes & Decorations
Tags
durchstosspunkt, ebene, gerade, mathematik, vektorgeometrie
Content image
Difficulty
(3, default)
Points
3 (default)
Language
GER (Deutsch)
Type
Calculative / Quantity
Creator uz
Decoration
File
Link