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: pmatrix x y z pmatrix pmatrix PEx PEy PEz pmatrix + u pmatrix ux uy uz pmatrix + v pmatrix vx vy vz pmatrix?

Solution:
tdplotsetmaincoords center tikzpicture>latex 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 --- nodeleft small bmx; drawcolorgreen!!black-> scaled cs --- noderight small bmy; drawcolorgreen!!black-> scaled cs --- nodeabove small bmz; drawdotted scaled cs PxXPyXPzX--PxXPyX; drawcolorblue scaled cs PxXPyXPzX--+-.*txX-.*tyX-.*tzX; drawcolorblue dashed scaled cs PxXPyXPzX--+.*txX.*tyX.*tzX; filldrawcoloryellow!!white fillyellow!!white opacity. scaled cs -.-CX--+.CX--.---.---cycle; filldrawcoloryellow!!white fillyellow!!white opacity. scaled cs -.---.--.---.----cycle; draw-> >stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidway left tiny vec vpmatrix tx ty tzpmatrix; draw-> >stealth colororange thick scaled cs PExXPEyXPEzX--+uxXuyXuzXnodemidway lefttiny vec a; draw-> >stealth colororange thick scaled cs PExXPEyXPEzX--+vxXvyXvzX nodemidway belowtiny vec b; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodeleft tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinates-- nodeleft below tiny D; tikzpicture center tdplotsetmaincoords center tikzpicture>latex 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 --- nodeleft small bmx; drawcolorgreen!!black-> scaled cs --- noderight small bmy; drawcolorgreen!!black-> scaled cs --- nodeabove small bmz; drawdotted scaled cs PxXPyXPzX--PxXPyX; drawcolorblue scaled cs PxXPyXPzX--+-.*txX-.*tyX-.*tzX; drawcolorblue dashed scaled cs PxXPyXPzX--+.*txX.*tyX.*tzX; filldrawcoloryellow!!white fillyellow!!white opacity. scaled cs -.-CX--+.CX--.---.---cycle; filldrawcoloryellow!!white fillyellow!!white opacity. scaled cs -.---.--.---.----cycle; draw-> >stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX nodemidway left tiny vec vpmatrix tx ty tzpmatrix; draw-> >stealth colororange thick scaled cs PExXPEyXPEzX--+uxXuyXuzXnodemidway lefttiny vec a; draw-> >stealth colororange thick scaled cs PExXPEyXPEzX--+vxXvyXvzX nodemidway belowtiny vec b; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesPxXPyXPzX nodeleft tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinates-- nodeleft below tiny D; tikzpicture center Der Normalenvektor auf der Ebene ist: pmatrix ux uy uz pmatrix times pmatrix vx vy vz pmatrix pmatrix NxX NyX NzX pmatrix Die Koordinatenform der Ebene sieht demnach so aus: mathcalE : NxX x + NyX y NzX z EihX Lösungsidee: bf einsetzen! center einsetzen in mathcalE cases x PxX txX t y PyX+tyX t z PzX tzX t cases center -PxX txX t+PyX+tyX t-PzX tzX t EihX -PxX txX t+PyX+tyX t-PzX tzX 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 PxX txX tX y PyX+tyX tX z PzX tzX tX cases center Das führt dann schliesslich auf den Durchstosspunkt D|-|-.