(%i1) "Cornelsen, S. 145/Nr. 46"$
(%i2) "a)"$
(%i3) "Drei Punkte A, B, C der Ebene"$
(%i4) A : transpose(matrix ([-20.0,30.0,-200.0]));
[ - 20.0 ]
[ ]
(%o4) [ 30.0 ]
[ ]
[ - 200.0 ]
(%i5) B : transpose(matrix ([120.0,180.0,-80.0]));
[ 120.0 ]
[ ]
(%o5) [ 180.0 ]
[ ]
[ - 80.0 ]
(%i6) C : transpose(matrix ([80.0,120.0,-120.0]));
[ 80.0 ]
[ ]
(%o6) [ 120.0 ]
[ ]
[ - 120.0 ]
(%i7) "Aufpunkt stuetzvec"$
(%i8) stuetzE : transpose(matrix ([-20.0,30.0,-200.0]));
[ - 20.0 ]
[ ]
(%o8) [ 30.0 ]
[ ]
[ - 200.0 ]
(%i9) "Richtungsvektoren uvec und vvec"$
(%i10) uvecE : B - A;
[ 140.0 ]
[ ]
(%o10) [ 150.0 ]
[ ]
[ 120.0 ]
(%i11) vvecE : C - A;
[ 100.0 ]
[ ]
(%o11) [ 90.0 ]
[ ]
[ 80.0 ]
(%i12) "Ebenengleichung"$
(%i13) display(stuetzE + r * uvecE + s * vvecE)$
[ - 20.0 ] [ 140.0 r ] [ 100.0 s ] [ 100.0 s + 140.0 r - 20.0 ]
[ ] [ ] [ ] [ ]
[ 30.0 ] + [ 150.0 r ] + [ 90.0 s ] = [ 90.0 s + 150.0 r + 30.0 ]
[ ] [ ] [ ] [ ]
[ - 200.0 ] [ 120.0 r ] [ 80.0 s ] [ 80.0 s + 120.0 r - 200.0 ]
(%i14) "Drei Punkte A, B, C der Ebene"$
(%i15) A : transpose(matrix ([-20.0,30.0,-236.0]));
[ - 20.0 ]
[ ]
(%o15) [ 30.0 ]
[ ]
[ - 236.0 ]
(%i16) B : transpose(matrix ([120.0,180.0,-120.0]));
[ 120.0 ]
[ ]
(%o16) [ 180.0 ]
[ ]
[ - 120.0 ]
(%i17) C : transpose(matrix ([80.0,120.0,-160.0]));
[ 80.0 ]
[ ]
(%o17) [ 120.0 ]
[ ]
[ - 160.0 ]
(%i18) "Aufpunkt stuetzvec"$
(%i19) stuetzE_ : transpose(matrix ([-20.0,30.0,-236.0]));
[ - 20.0 ]
[ ]
(%o19) [ 30.0 ]
[ ]
[ - 236.0 ]
(%i20) "Richtungsvektoren uvec und vvec"$
(%i21) uvecE_ : B - A;
[ 140.0 ]
[ ]
(%o21) [ 150.0 ]
[ ]
[ 116.0 ]
(%i22) vvecE_ : C - A;
[ 100.0 ]
[ ]
(%o22) [ 90.0 ]
[ ]
[ 76.0 ]
(%i23) "Ebenengleichung"$
(%i24) display(stuetzE_ + r * uvecE_ + s * vvecE_)$
[ - 20.0 ] [ 140.0 r ] [ 100.0 s ] [ 100.0 s + 140.0 r - 20.0 ]
[ ] [ ] [ ] [ ]
[ 30.0 ] + [ 150.0 r ] + [ 90.0 s ] = [ 90.0 s + 150.0 r + 30.0 ]
[ ] [ ] [ ] [ ]
[ - 236.0 ] [ 116.0 r ] [ 76.0 s ] [ 76.0 s + 116.0 r - 236.0 ]
(%i25) "b)"$
(%i26) "Schnittgerade der Ebenen E und E_"$
(%i27) display(transpose(matrix([0.0,600.000000000000, -0.000000000000])) + r * transpose(matrix([2.0,3.0,2.0])))$
[ 0.0 ] [ 2.0 r ] [ 2.0 r ]
[ ] [ ] [ ]
[ 600.0 ] + [ 3.0 r ] = [ 3.0 r + 600.0 ]
[ ] [ ] [ ]
[ -0.0 ] [ 2.0 r ] [ 2.0 r ]
(%i28) "Schnittwinkel alpha = 5.885129333686"$
(%i29) "c)"$
(%i30) "x-Vektor xvec"$
(%i31) xvec : transpose(matrix ([x,y,z]));
[ x ]
[ ]
(%o31) [ y ]
[ ]
[ z ]
(%i32) "Aufpunkt stuetzvec"$
(%i33) stuetzg_t : transpose(matrix ([-200.0,200.0,0.0]));
[ - 200.0 ]
[ ]
(%o33) [ 200.0 ]
[ ]
[ 0.0 ]
(%i34) "Richtungsvektor uvec"$
(%i35) uvecg_t : transpose(matrix ([2.0,-2.0,-1.0]));
[ 2.0 ]
[ ]
(%o35) [ - 2.0 ]
[ ]
[ - 1.0 ]
(%i36) "Geradengleichung g"$
(%i37) display(stuetzg_t + r * uvecg_t)$
[ - 200.0 ] [ 2.0 r ] [ 2.0 r - 200.0 ]
[ ] [ ] [ ]
[ 200.0 ] + [ - 2.0 r ] = [ 200.0 - 2.0 r ]
[ ] [ ] [ ]
[ 0.0 ] [ - 1.0 r ] [ - 1.0 r ]
(%i38) g_t : stuetzg_t + r * uvecg_t$
(%i39) "Schnittpunkt der Gerade g_t und der Ebene E"$
(%i40) sp : transpose(matrix([150.00000000000000,-150.00000000000000,-175.00000000000000]));
[ 150.0 ]
[ ]
(%o40) [ - 150.0 ]
[ ]
[ - 175.0 ]
(%i41) "Schnittwinkel alpha = 22.392687805402"$
(%i42) "Schnittpunkt der Gerade g_t und der Ebene E_"$
(%i43) sp : transpose(matrix([280.00000000000000,-280.00000000000000,-240.00000000000000]));
[ 280.0 ]
[ ]
(%o43) [ - 280.0 ]
[ ]
[ - 240.0 ]
(%i44) "Schnittwinkel alpha = 16.865704761298"$
(%i45) "Vektor v_diff"$
(%i46) v_diff: transpose(matrix ([150.0,-150.0,-175.0])) - transpose(matrix ([-200.0,200.0,0.0]));
[ 350.0 ]
[ ]
(%o46) [ - 350.0 ]
[ ]
[ - 175.0 ]
(%i47) "Laenge l des Vektors v_diff: l = 525.0"$
(%i48) "d)"$
(%i49) "x-Vektor xvec"$
(%i50) xvec : transpose(matrix ([x,y,z]));
[ x ]
[ ]
(%o50) [ y ]
[ ]
[ z ]
(%i51) "Aufpunkt stuetzvec"$
(%i52) stuetzg_t2 : transpose(matrix ([-100.0,450.0,0.0]));
[ - 100.0 ]
[ ]
(%o52) [ 450.0 ]
[ ]
[ 0.0 ]
(%i53) "Richtungsvektor uvec"$
(%i54) uvecg_t2 : transpose(matrix ([0.0,0.0,1.0]));
[ 0.0 ]
[ ]
(%o54) [ 0.0 ]
[ ]
[ 1.0 ]
(%i55) "Geradengleichung g"$
(%i56) display(stuetzg_t2 + r * uvecg_t2)$
[ - 100.0 ] [ 0.0 ] [ - 100.0 ]
[ ] [ ] [ ]
[ 450.0 ] + [ 0.0 ] = [ 450.0 ]
[ ] [ ] [ ]
[ 0.0 ] [ 1.0 r ] [ 1.0 r ]
(%i57) g_t2 : stuetzg_t2 + r * uvecg_t2$
(%i58) "Schnittpunkt der Gerade g_t2 und der Ebene E"$
(%i59) sp : transpose(matrix([-100.00000000000000,450.00000000000000,-100.00000000000000]));
[ - 100.0 ]
[ ]
(%o59) [ 450.0 ]
[ ]
[ - 100.0 ]
(%i60) "Schnittwinkel alpha = 58.997280866126"$
(%i61) "Punkt SPEg_t2 liegt auf der Gerade gEE_"$
(%i62) Antwort: die senkrechte Bohrung trifft die Gerade g aus b), in der das Kohlefloez endet.