Abstand von Punkt zu Gerade
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Exercise:
Welchen Abstand hat der Punkt PPx|Py|Pz zu der Geraden g: pmatrix x y z pmatrix pmatrix gx gy gz pmatrix + t pmatrix tx ty tz pmatrix?
Solution:
tikzset glow/.style preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. bf . Berechnungsmethode 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 LxX LyX LzX--LxX LyX ; drawdotted scaled cs PxX PyX PzX--PxX PyX ; drawdotted scaled cs gxX gyX gzX--gxX gyX ; drawcolorblue scaled cs gxXgyXgzX--+-*txX-*tyX-*tzX nodeleftsmallg; drawcolorblue scaled cs gxXgyXgzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs gxXgyXgzX--+txXtyXtzX nodeleft tiny pmatrix tx ty tzpmatrix vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxXgyXgzX noderight tiny Dgx|gy|gz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillviolet!!white coordinatesLxXLyXLzX nodeleftred!!blue tiny L; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!yellow coordinatesPxXPyXPzX noderight red!!yellow tiny PPx|Py|Pz; draw- stealth colorred!!yellow thick dashed scaled cs --+PxXPyXPzX nodemidway above tiny vec r_P; draw- stealth colorred!!blue thick dashed scaled cs --+LxXLyXLzX nodemidway left tiny vec r_L; draw- stealth colorolive thick scaled cs PxXPyXPzX--LxX LyX LzX nodemidway above tiny vec d; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesLxXLyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesgxXgyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesPxXPyX; tikzpicture center Um den Abstand zu berechnen muss als erstes der Lotfusspunkt L auf der Geraden g gefunden werden. Weil der Vektor vecPL rechtwinklig zum Vektor vec v in Richtung der Geraden sein muss deren Skalarprodukt also verschwinden muss gilt: leftvec L - vec Pright vec v &mustbe leftpmatrix L_x L_y L_zpmatrix - pmatrix Px Py Pz pmatrix right pmatrix tx ty tz pmatrix &mustbe Der Punkt L ist unbekannt aber liegt auf der Geraden weshalb man schreiben kann: leftpmatrix gx - t tx gy - t ty gz - t tzpmatrix - pmatrix Px Py Pz pmatrix right pmatrix tx ty tz pmatrix &mustbe pmatrix - t tx - - t ty - - t tzpmatrix pmatrix tx ty tz pmatrix &mustbe Skalarprodukt ausrechnen und auflösen nach t liefert: t . Damit findet man den Lotfusspunkt: pmatrix L_x L_y L_zpmatrix pmatrix gx gy gz pmatrix + . pmatrix tx ty tz pmatrix pmatrix Lx Ly Lz pmatrix Der Abstand zwischen den Punkten L und P kann nun leicht berechnet werden: vec d vec r_L - vec r_P pmatrix x_L y_L z_L pmatrix - pmatrix x_P y_P z_P pmatrix pmatrix LxX LyX LzX pmatrix - pmatrix PxX PyX PzX pmatrix pmatrix AX BX CX pmatrix |vec d| sqrtx_L-x_P^+y_L-y_P^+z_L-z_P^ sqrtLxX-PxX^+LyX-PyX^+LzX-PzX^ sqrtAX^+BX^+CX^ G bf . Berechnungsmethode 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 PxX PyX PzX--PxX PyX ; drawdotted scaled cs gxX gyX gzX--gxX gyX ; drawcolorblue scaled cs gxXgyXgzX--+-*txX-*tyX-*tzX nodeleftsmallg; drawcolorblue scaled cs gxXgyXgzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs gxXgyXgzX--+txXtyXtzX nodemidwayleft tiny vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxX+txXgyX+tyXgzX+tzX nodeleft tiny BBxX|ByX|BzX; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxXgyXgzX noderight tiny Agx|gy|gz; draw- stealth colorred!!white dashed thick scaled cs glowred PxXPyXPzX--+txXtyXtzX nodemidway right tiny vec v; draw- stealth colororange!!white dashed thick scaled cs glowyellow gxXgyXgzX--PxXPyXPzX nodemidway below tiny vec AP; draw- stealth colorred!!white dashed thick scaled cs glowyellow BxXByXBzX--+sxXsyXszX; filldrawcolorblack fillyellow!!white opacity. scaled csgxXgyXgzX--BxXByXBzX--QxXQyXQzX--PxXPyXPzX--cycle; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!yellow coordinatesPxXPyXPzX noderight red!!yellow tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblack coordinatesQxXQyXQzX noderight tiny QQxX|QyX|QzX; draw- stealth colorolive thick scaled cs PxXPyXPzX--LxX LyX LzX nodemidway above tiny vec d; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesLxXLyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesgxXgyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesPxXPyX; tikzpicture center Die Fläche des von AB und P aufgespannten Parallelogramms lässt sich darstellen als: F_APQB overlineAB |vec d| Diese Gleichung kann man mithilfe des Kreuzprodukts umformulieren in: F_APQB |vecABtimes vecAP| |vec d| d fracF_APQBoverlineAB d frac|vecABtimes vecAP|overlineAB In diese Formel können nun die konkreten Werte der Aufgabe eingesetzt werden: d fracleft|pmatrix bxX byX bzX pmatrix times pmatrix pxX pyX pzX pmatrixright| overlinepmatrix bxX byX bzX pmatrix fracleft|pmatrix kxX kyX kzX pmatrixright|overlinepmatrix bxX byX bzX pmatrix fracDE F bf . Berechnungsmethode 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 LxX LyX LzX--LxX LyX ; drawdotted scaled cs PxX PyX PzX--PxX PyX ; drawdotted scaled cs gxX gyX gzX--gxX gyX ; drawcolorblue scaled cs gxXgyXgzX--+-*txX-*tyX-*tzX nodeleftsmallg; drawcolorblue scaled cs gxXgyXgzX--+*txX*tyX*tzX; drawcoloryellow!!red very thick scaled cs glowyellow --.. nodebelowsmallmathcalE; draw- stealth colorred thick scaled cs gxXgyXgzX--+txXtyXtzX nodeleft tiny pmatrix tx ty tzpmatrix vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxXgyXgzX noderight tiny Dgx|gy|gz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillviolet!!white coordinatesLxXLyXLzX nodeleftred!!blue tiny SLx|Ly|Lz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!yellow coordinatesPxXPyXPzX noderight red!!yellow tiny PPx|Py|Pz; draw- stealth colorred!!yellow thick dashed scaled cs --+PxXPyXPzX nodemidway above tiny vec r_P; draw- stealth colorolive thick scaled cs PxXPyXPzX--LxX LyX LzX nodemidway above tiny vec d; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesLxXLyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesgxXgyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesPxXPyX; tikzpicture center Normalenvektor von g ablesen: vec n pmatrix tx ty tz pmatrix tx x + ty y + tz z + D P einsetzen und so D bestimmen: hxX+hyX+hzX + D Rightarrow D HX Daraus folgt dann die Ebenengleichung: tx x + tz z HX Normalebene durch P zu g schneiden mit g dafür werden zunächst die Komponentengleichungen von g aufgestellt: center x gx+tx t y gy+ty t z gz+tz t center Anschliess wird der Schnittpunkt S bestimmt: center tx gx+tx t + tz gz+tz t HX ixX+jxX t + izX+jzX t HX lX+mX t t nX Rightarrow S uxX|uyX|uzX center Vergleicht man diesen S mit dem Lotfusspunkt L aus der ersten Berechungsmethode fällt auf dass es sich bei den beiden um den gleichen Punkt handelt. Nun muss nur noch der Abstand zwischen P und S bestimmt werden: vec d vec r_S - vec r_P pmatrix x_S y_S z_S pmatrix - pmatrix x_P y_P z_P pmatrix pmatrix uxX uyX uzX pmatrix - pmatrix PxX PyX PzX pmatrix pmatrix AX BX CX pmatrix |vec d| sqrtx_S-x_P^+y_S-y_P^+z_S-z_P^ sqrtLxX-PxX^+LyX-PyX^+LzX-PzX^ sqrtAX^+BX^+CX^ G
Welchen Abstand hat der Punkt PPx|Py|Pz zu der Geraden g: pmatrix x y z pmatrix pmatrix gx gy gz pmatrix + t pmatrix tx ty tz pmatrix?
Solution:
tikzset glow/.style preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. preaction# draw line joinround line width.pt opacity. bf . Berechnungsmethode 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 LxX LyX LzX--LxX LyX ; drawdotted scaled cs PxX PyX PzX--PxX PyX ; drawdotted scaled cs gxX gyX gzX--gxX gyX ; drawcolorblue scaled cs gxXgyXgzX--+-*txX-*tyX-*tzX nodeleftsmallg; drawcolorblue scaled cs gxXgyXgzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs gxXgyXgzX--+txXtyXtzX nodeleft tiny pmatrix tx ty tzpmatrix vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxXgyXgzX noderight tiny Dgx|gy|gz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillviolet!!white coordinatesLxXLyXLzX nodeleftred!!blue tiny L; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!yellow coordinatesPxXPyXPzX noderight red!!yellow tiny PPx|Py|Pz; draw- stealth colorred!!yellow thick dashed scaled cs --+PxXPyXPzX nodemidway above tiny vec r_P; draw- stealth colorred!!blue thick dashed scaled cs --+LxXLyXLzX nodemidway left tiny vec r_L; draw- stealth colorolive thick scaled cs PxXPyXPzX--LxX LyX LzX nodemidway above tiny vec d; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesLxXLyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesgxXgyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesPxXPyX; tikzpicture center Um den Abstand zu berechnen muss als erstes der Lotfusspunkt L auf der Geraden g gefunden werden. Weil der Vektor vecPL rechtwinklig zum Vektor vec v in Richtung der Geraden sein muss deren Skalarprodukt also verschwinden muss gilt: leftvec L - vec Pright vec v &mustbe leftpmatrix L_x L_y L_zpmatrix - pmatrix Px Py Pz pmatrix right pmatrix tx ty tz pmatrix &mustbe Der Punkt L ist unbekannt aber liegt auf der Geraden weshalb man schreiben kann: leftpmatrix gx - t tx gy - t ty gz - t tzpmatrix - pmatrix Px Py Pz pmatrix right pmatrix tx ty tz pmatrix &mustbe pmatrix - t tx - - t ty - - t tzpmatrix pmatrix tx ty tz pmatrix &mustbe Skalarprodukt ausrechnen und auflösen nach t liefert: t . Damit findet man den Lotfusspunkt: pmatrix L_x L_y L_zpmatrix pmatrix gx gy gz pmatrix + . pmatrix tx ty tz pmatrix pmatrix Lx Ly Lz pmatrix Der Abstand zwischen den Punkten L und P kann nun leicht berechnet werden: vec d vec r_L - vec r_P pmatrix x_L y_L z_L pmatrix - pmatrix x_P y_P z_P pmatrix pmatrix LxX LyX LzX pmatrix - pmatrix PxX PyX PzX pmatrix pmatrix AX BX CX pmatrix |vec d| sqrtx_L-x_P^+y_L-y_P^+z_L-z_P^ sqrtLxX-PxX^+LyX-PyX^+LzX-PzX^ sqrtAX^+BX^+CX^ G bf . Berechnungsmethode 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 PxX PyX PzX--PxX PyX ; drawdotted scaled cs gxX gyX gzX--gxX gyX ; drawcolorblue scaled cs gxXgyXgzX--+-*txX-*tyX-*tzX nodeleftsmallg; drawcolorblue scaled cs gxXgyXgzX--+*txX*tyX*tzX; draw- stealth colorred thick scaled cs gxXgyXgzX--+txXtyXtzX nodemidwayleft tiny vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxX+txXgyX+tyXgzX+tzX nodeleft tiny BBxX|ByX|BzX; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxXgyXgzX noderight tiny Agx|gy|gz; draw- stealth colorred!!white dashed thick scaled cs glowred PxXPyXPzX--+txXtyXtzX nodemidway right tiny vec v; draw- stealth colororange!!white dashed thick scaled cs glowyellow gxXgyXgzX--PxXPyXPzX nodemidway below tiny vec AP; draw- stealth colorred!!white dashed thick scaled cs glowyellow BxXByXBzX--+sxXsyXszX; filldrawcolorblack fillyellow!!white opacity. scaled csgxXgyXgzX--BxXByXBzX--QxXQyXQzX--PxXPyXPzX--cycle; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!yellow coordinatesPxXPyXPzX noderight red!!yellow tiny PPx|Py|Pz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblack coordinatesQxXQyXQzX noderight tiny QQxX|QyX|QzX; draw- stealth colorolive thick scaled cs PxXPyXPzX--LxX LyX LzX nodemidway above tiny vec d; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesLxXLyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesgxXgyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesPxXPyX; tikzpicture center Die Fläche des von AB und P aufgespannten Parallelogramms lässt sich darstellen als: F_APQB overlineAB |vec d| Diese Gleichung kann man mithilfe des Kreuzprodukts umformulieren in: F_APQB |vecABtimes vecAP| |vec d| d fracF_APQBoverlineAB d frac|vecABtimes vecAP|overlineAB In diese Formel können nun die konkreten Werte der Aufgabe eingesetzt werden: d fracleft|pmatrix bxX byX bzX pmatrix times pmatrix pxX pyX pzX pmatrixright| overlinepmatrix bxX byX bzX pmatrix fracleft|pmatrix kxX kyX kzX pmatrixright|overlinepmatrix bxX byX bzX pmatrix fracDE F bf . Berechnungsmethode 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 LxX LyX LzX--LxX LyX ; drawdotted scaled cs PxX PyX PzX--PxX PyX ; drawdotted scaled cs gxX gyX gzX--gxX gyX ; drawcolorblue scaled cs gxXgyXgzX--+-*txX-*tyX-*tzX nodeleftsmallg; drawcolorblue scaled cs gxXgyXgzX--+*txX*tyX*tzX; drawcoloryellow!!red very thick scaled cs glowyellow --.. nodebelowsmallmathcalE; draw- stealth colorred thick scaled cs gxXgyXgzX--+txXtyXtzX nodeleft tiny pmatrix tx ty tzpmatrix vec v; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillblue!!white coordinatesgxXgyXgzX noderight tiny Dgx|gy|gz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillviolet!!white coordinatesLxXLyXLzX nodeleftred!!blue tiny SLx|Ly|Lz; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!yellow coordinatesPxXPyXPzX noderight red!!yellow tiny PPx|Py|Pz; draw- stealth colorred!!yellow thick dashed scaled cs --+PxXPyXPzX nodemidway above tiny vec r_P; draw- stealth colorolive thick scaled cs PxXPyXPzX--LxX LyX LzX nodemidway above tiny vec d; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesLxXLyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesgxXgyX; shadedrawscaled cs plot only marks mark* mark size.pt mark optionsfillblack coordinatesPxXPyX; tikzpicture center Normalenvektor von g ablesen: vec n pmatrix tx ty tz pmatrix tx x + ty y + tz z + D P einsetzen und so D bestimmen: hxX+hyX+hzX + D Rightarrow D HX Daraus folgt dann die Ebenengleichung: tx x + tz z HX Normalebene durch P zu g schneiden mit g dafür werden zunächst die Komponentengleichungen von g aufgestellt: center x gx+tx t y gy+ty t z gz+tz t center Anschliess wird der Schnittpunkt S bestimmt: center tx gx+tx t + tz gz+tz t HX ixX+jxX t + izX+jzX t HX lX+mX t t nX Rightarrow S uxX|uyX|uzX center Vergleicht man diesen S mit dem Lotfusspunkt L aus der ersten Berechungsmethode fällt auf dass es sich bei den beiden um den gleichen Punkt handelt. Nun muss nur noch der Abstand zwischen P und S bestimmt werden: vec d vec r_S - vec r_P pmatrix x_S y_S z_S pmatrix - pmatrix x_P y_P z_P pmatrix pmatrix uxX uyX uzX pmatrix - pmatrix PxX PyX PzX pmatrix pmatrix AX BX CX pmatrix |vec d| sqrtx_S-x_P^+y_S-y_P^+z_S-z_P^ sqrtLxX-PxX^+LyX-PyX^+LzX-PzX^ sqrtAX^+BX^+CX^ G