Exercise:
Wie gross ist der Winkel zwischen der Geraden g: pmatrix x y z pmatrix pmatrix Px Py Pz pmatrix + t pmatrix tx ty tz pmatrix und der xy-Ebene?

Solution:
tdplotsetmaincoords 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. makeatletter tikzoptioncanvas is plane@setOxy# def@setOxy O###x###y###% deftikz@plane@originpgfpoxyz###% deftikz@plane@xpgfpoxyz###% deftikz@plane@ypgfpoxyz###% tikz@canvas@is@plane makeatother 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 --- nodeabove small bmx; drawcolorgreen!!black- scaled cs --- nodeabove small bmy; drawcolorgreen!!black- scaled cs --- nodeabove small bmz; scopecanvas is planeO-.x-.y-.blue! drawthick -.. arc ::. node scale. midway left alpha; scope filldrawcolorblack fillyellow!!white opacity. scaled cs glowyellow ------------cycle; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX noderightsmallg; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX noderight tiny vec vpmatrix tx ty tzpmatrix; draw- stealth colorred!!blue thick scaled cs --+PxXPyXPzX nodemidwayright tiny vec r_P; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!blue coordinatesPxXPyXPzX nodeleftred!!blue tiny P; tikzpicture center tdplotsetmaincoords 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. makeatletter tikzoptioncanvas is plane@setOxy# def@setOxy O###x###y###% deftikz@plane@originpgfpoxyz###% deftikz@plane@xpgfpoxyz###% deftikz@plane@ypgfpoxyz###% tikz@canvas@is@plane makeatother 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 --- nodebelow small bmx; drawcolorgreen!!black- scaled cs --- nodebelow small bmy; drawcolorgreen!!black- scaled cs --- nodeabove small bmz; filldrawcolorblack fillyellow!!white opacity. scaled cs glowyellow ------------cycle; drawcolorblue scaled cs PxXPyXPzX--+-*txX-*tyX-*tzX; drawcolorblue scaled cs PxXPyXPzX--+*txX*tyX*tzX nodebelowsmallg; draw- stealth colorred thick scaled cs PxXPyXPzX--+txXtyXtzX noderight tiny vec vpmatrix tx ty tzpmatrix; draw- stealth colorred!!blue thick scaled cs --+PxXPyXPzX nodemidwayright tiny vec r_P; shadedrawscaled cs plot only marks mark* mark sizept mark optionsfillred!!blue coordinatesPxXPyXPzX nodeleftred!!blue tiny P; tikzpicture center Zunächst muss der Normalenvektor der Ebene bestimmt werden was in diesem Fall sehr simpel ist da es sich ja um die xy-Ebene handelt: vec npmatrix nx ny nzpmatrix Nun kann mit diesem Normalenvektor vec n und dem Richtungsvektor vec a der Gerade g mithilfe des Skalarproduktes der gesuchte Winkel berechnet werden: sinalpha fracveca vecn|veca| |vecn| fracpmatrix txX tyX tzXpmatrix pmatrix nxX nyX nzXpmatrixleft|pmatrix txX tyX tzXpmatrixright| left|pmatrix nxX nyX nzXpmatrixright| fracmxX+myX+mzXsqrttxX^+tyX^+tzX^ sqrtnxX^+nyX^+nzX^ fracMXsqrtOX sqrtPX alpha arcsinfracMXsqrtOX sqrtPX arcsinfracMXsqrt arcsinfracsqrt ang.