(%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"$