(%i1) "Cornelsen, S. 151/Nr. 5"$
(%i2) "a)"$
(%i3) "Ebene E: B, R, P"$
(%i4) "Drei Punkte A, B, C der Ebene"$
(%i5) A : transpose(matrix ([4.0,4.0,0.0]));
[ 4.0 ]
[ ]
(%o5) [ 4.0 ]
[ ]
[ 0.0 ]
(%i6) B : transpose(matrix ([4.0,0.0,4.0]));
[ 4.0 ]
[ ]
(%o6) [ 0.0 ]
[ ]
[ 4.0 ]
(%i7) C : transpose(matrix ([0.0,4.0,4.0]));
[ 0.0 ]
[ ]
(%o7) [ 4.0 ]
[ ]
[ 4.0 ]
(%i8) "Aufpunkt stuetzvec"$
(%i9) stuetzE1 : transpose(matrix ([4.0,4.0,0.0]));
[ 4.0 ]
[ ]
(%o9) [ 4.0 ]
[ ]
[ 0.0 ]
(%i10) "Richtungsvektoren uvec und vvec"$
(%i11) uvecE1 : B - A;
[ 0.0 ]
[ ]
(%o11) [ - 4.0 ]
[ ]
[ 4.0 ]
(%i12) vvecE1 : C - A;
[ - 4.0 ]
[ ]
(%o12) [ 0.0 ]
[ ]
[ 4.0 ]
(%i13) "Ebenengleichung"$
(%i14) display(stuetzE1 + r * uvecE1 + s * vvecE1)$
[ 4.0 ] [ 0.0 ] [ - 4.0 s ] [ 4.0 - 4.0 s ]
[ ] [ ] [ ] [ ]
[ 4.0 ] + [ - 4.0 r ] + [ 0.0 ] = [ 4.0 - 4.0 r ]
[ ] [ ] [ ] [ ]
[ 0.0 ] [ 4.0 r ] [ 4.0 s ] [ 4.0 s + 4.0 r ]
(%i31) "Normalenvektor nvecE1"$
(%i32) nvecE1 : transpose(coefmatrix([detE1],[x,y,z]));
[ - 16 ]
[ ]
(%o32) [ - 16 ]
[ ]
[ - 16 ]
(%i33) dE1 : (-1) *determinant( col(augcoefmatrix([detE1],[x,y,z]),4) );
(%o33) - 128.0
(%i34) koordinatenform(x,n,d) := (print( x . n = d) )$
(%i35) "Koordinatenform"$
(%i36) koordinatenform(xvec,nvecE1,dE1)$
- 16 z - 16 y - 16 x = - 128.0
<=> x + y + z = 8
(%i37) "Abstand der Ebene E1 vom Ursprung: d = 4.618802153517"$
(%i38) "b)"$
(%i39) "Zwei Punkte A, B der Gerade g"$
(%i40) A : transpose(matrix ([0.0,0.0,0.0]));
[ 0.0 ]
[ ]
(%o40) [ 0.0 ]
[ ]
[ 0.0 ]
(%i41) B : transpose(matrix ([4.0,4.0,4.0]));
[ 4.0 ]
[ ]
(%o41) [ 4.0 ]
[ ]
[ 4.0 ]
(%i42) "Aufpunkt stuetzvec"$
(%i43) stuetzg_DQ : transpose(matrix ([0.0,0.0,0.0]));
[ 0.0 ]
[ ]
(%o43) [ 0.0 ]
[ ]
[ 0.0 ]
(%i44) "Richtungsvektor uvec"$
(%i45) uvecg_DQ : B - A;
[ 4.0 ]
[ ]
(%o45) [ 4.0 ]
[ ]
[ 4.0 ]
(%i46) "x-Vektor xvec"$
(%i47) xvec : transpose(matrix ([x,y,z]));
[ x ]
[ ]
(%o47) [ y ]
[ ]
[ z ]
(%i48) "Schnittpunkt der Gerade g_DQ und der Ebene E1"$
(%i49) sp : transpose(matrix([ 2.66666666666667, 2.66666666666667, 2.66666666666667]));
[ 2.66666666666667 ]
[ ]
(%o49) [ 2.66666666666667 ]
[ ]
[ 2.66666666666667 ]
(%i50) "Vektor v1"$
(%i51) v1: transpose(matrix ([2.66666666667,2.66666666667,2.66666666667])) - transpose(matrix ([0.0,0.0,0.0]));
[ 2.66666666667 ]
[ ]
(%o51) [ 2.66666666667 ]
[ ]
[ 2.66666666667 ]
(%i52) "Laenge l des Vektors v1: l = 4.61880215352"$
(%i53) "DS = 4.6188"$
(%i54) "Vektor v2"$
(%i55) v2: transpose(matrix ([4.0,4.0,4.0])) - transpose(matrix ([0.0,0.0,0.0]));
[ 4.0 ]
[ ]
(%o55) [ 4.0 ]
[ ]
[ 4.0 ]
(%i56) "Laenge l des Vektors v2: l = 6.92820323028"$
(%i57) "DQ = 6,9282 => |DS| > |SQ|"$
(%i58) "c)"$
(%i59) "Ebene F durch M, C, R"$
(%i60) "Drei Punkte A, B, C der Ebene"$
(%i61) A : transpose(matrix ([0.0,4.0,0.0]));
[ 0.0 ]
[ ]
(%o61) [ 4.0 ]
[ ]
[ 0.0 ]
(%i62) B : transpose(matrix ([0.0,4.0,4.0]));
[ 0.0 ]
[ ]
(%o62) [ 4.0 ]
[ ]
[ 4.0 ]
(%i63) C : transpose(matrix ([4.0,2.0,0.0]));
[ 4.0 ]
[ ]
(%o63) [ 2.0 ]
[ ]
[ 0.0 ]
(%i64) "Aufpunkt stuetzvec"$
(%i65) stuetzE2 : transpose(matrix ([0.0,4.0,0.0]));
[ 0.0 ]
[ ]
(%o65) [ 4.0 ]
[ ]
[ 0.0 ]
(%i66) "Richtungsvektoren uvec und vvec"$
(%i67) uvecE2 : B - A;
[ 0.0 ]
[ ]
(%o67) [ 0.0 ]
[ ]
[ 4.0 ]
(%i68) vvecE2 : C - A;
[ 4.0 ]
[ ]
(%o68) [ - 2.0 ]
[ ]
[ 0.0 ]
(%i69) "Ebenengleichung"$
(%i70) display(stuetzE2 + r * uvecE2 + s * vvecE2)$
[ 0.0 ] [ 0.0 ] [ 4.0 s ] [ 4.0 s ]
[ ] [ ] [ ] [ ]
[ 4.0 ] + [ 0.0 ] + [ - 2.0 s ] = [ 4.0 - 2.0 s ]
[ ] [ ] [ ] [ ]
[ 0.0 ] [ 4.0 r ] [ 0.0 ] [ 4.0 r ]
(%i71) "Schnittgerade der Ebenen E1 und E2"$
(%i72) display(transpose(matrix([0.0, 4.000000000000, 4.000000000000])) + r * transpose(matrix([256.0,-128.0,-128.0])))$
[ 0.0 ] [ 256.0 r ] [ 256.0 r ]
[ ] [ ] [ ]
[ 4.0 ] + [ - 128.0 r ] = [ 4.0 - 128.0 r ]
[ ] [ ] [ ]
[ 4.0 ] [ - 128.0 r ] [ 4.0 - 128.0 r ]
(%i73) "Schnittwinkel alpha = 39.231520483592"$
(%i74) "g_s: [0+2r ; 4-r ; 4-r]"$
(%i75) "d)"$
(%i76) "Quader (Punkte A, B, C, D, E)"$
(%i77) Qu1_A : transpose(matrix ([0.0,0.0,0.0]));
[ 0.0 ]
[ ]
(%o77) [ 0.0 ]
[ ]
[ 0.0 ]
(%i78) Qu1_B : transpose(matrix ([4.0,0.0,0.0]));
[ 4.0 ]
[ ]
(%o78) [ 0.0 ]
[ ]
[ 0.0 ]
(%i79) Qu1_C : transpose(matrix ([4.0,4.0,0.0]));
[ 4.0 ]
[ ]
(%o79) [ 4.0 ]
[ ]
[ 0.0 ]
(%i80) Qu1_D : transpose(matrix ([0.0,4.0,0.0]));
[ 0.0 ]
[ ]
(%o80) [ 4.0 ]
[ ]
[ 0.0 ]
(%i81) Qu1_E : transpose(matrix ([0.0,0.0,4.0]));
[ 0.0 ]
[ ]
(%o81) [ 0.0 ]
[ ]
[ 4.0 ]
(%i82) Qu1_F : transpose(matrix ([4.0,0.0,4.0]));
[ 4.0 ]
[ ]
(%o82) [ 0.0 ]
[ ]
[ 4.0 ]
(%i83) Qu1_G : transpose(matrix ([4.0,4.0,4.0]));
[ 4.0 ]
[ ]
(%o83) [ 4.0 ]
[ ]
[ 4.0 ]
(%i84) Qu1_H : transpose(matrix ([0.0,4.0,4.0]));
[ 0.0 ]
[ ]
(%o84) [ 4.0 ]
[ ]
[ 4.0 ]
(%i85) "Zwei Punkte A, B der Gerade g"$
(%i86) A : transpose(matrix ([0.0,0.0,6.0]));
[ 0.0 ]
[ ]
(%o86) [ 0.0 ]
[ ]
[ 6.0 ]
(%i87) B : transpose(matrix ([1.5,6.0,0.0]));
[ 1.5 ]
[ ]
(%o87) [ 6.0 ]
[ ]
[ 0.0 ]
(%i88) "Aufpunkt stuetzvec"$
(%i89) stuetzg_UV : transpose(matrix ([0.0,0.0,6.0]));
[ 0.0 ]
[ ]
(%o89) [ 0.0 ]
[ ]
[ 6.0 ]
(%i90) "Richtungsvektor uvec"$
(%i91) uvecg_UV : B - A;
[ 1.5 ]
[ ]
(%o91) [ 6.0 ]
[ ]
[ - 6.0 ]
(%i92) "x-Vektor xvec"$
(%i93) xvec : transpose(matrix ([x,y,z]));
[ x ]
[ ]
(%o93) [ y ]
[ ]
[ z ]
(%i94) "Schnittpunkt der Gerade g_UV und der Ebene E1"$
(%i95) sp : transpose(matrix([ 2.00000000000000, 8.00000000000000, -2.00000000000000]));
[ 2.0 ]
[ ]
(%o95) [ 8.0 ]
[ ]
[ - 2.0 ]
(%i96) "Schnittwinkel alpha = 5.768181186188"$
(%i97) "Schnittpunkt mit Ebene E liegt ausserhalb des "$
(%i98) "Wuerfels, also trifft g_UV nicht das Dreieck BRP!"$
(%i99) "Der Strahl trifft den Wuerfel in der Ebene ABP, bzw. PQR"$