Exercise
https://texercises.com/exercise/gerade-durch-parametrisierte-ebene
Question
Solution
Short
Video
No explanation / solution video for this exercise has yet been created.

Visit our YouTube-Channel to see solutions for other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Report An Error
You are on texercises.com.
reCaptcha will only work on our main-domain \(\TeX\)ercises.com!
Meta Information
\(\LaTeX\)-Code
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|-|-.
Contained in these collections:

Attributes & Decorations
Tags durchstosspunkt, ebene, gerade, mathematik, vektorgeometrie
Difficulty
(3, default)
Points 3 (default)
Language GER (Deutsch)
Type Calculative / Quantity
Creator uz
Decoration