Mathematik Q2: 7. Mai 2015


(%i1) "Cornelsen, S. 143/Nr. 39a)"$

(%i2) "Drei Punkte A, B, C der Ebene"$

(%i3) A : transpose(matrix ([6.0,2.0,6.0]));
                                    [ 6.0 ]
                                    [     ]
(%o3)                               [ 2.0 ]
                                    [     ]
                                    [ 6.0 ]

(%i4) B : transpose(matrix ([6.0,6.0,2.0]));
                                    [ 6.0 ]
                                    [     ]
(%o4)                               [ 6.0 ]
                                    [     ]
                                    [ 2.0 ]

(%i5) C : transpose(matrix ([3.0,6.0,6.0]));
                                    [ 3.0 ]
                                    [     ]
(%o5)                               [ 6.0 ]
                                    [     ]
                                    [ 6.0 ]

(%i6) "Aufpunkt stuetzvec"$

(%i7) stuetzE1 : transpose(matrix ([6.0,2.0,6.0]));
                                    [ 6.0 ]
                                    [     ]
(%o7)                               [ 2.0 ]
                                    [     ]
                                    [ 6.0 ]

(%i8) "Richtungsvektoren uvec und vvec"$

(%i9) uvecE1 : B - A;
                                   [  0.0  ]
                                   [       ]
(%o9)                              [  4.0  ]
                                   [       ]
                                   [ - 4.0 ]

(%i10) vvecE1 : C - A;
                                   [ - 3.0 ]
                                   [       ]
(%o10)                             [  4.0  ]
                                   [       ]
                                   [  0.0  ]

(%i11) "Ebenengleichung"$

(%i12) display(stuetzE1 +  r * uvecE1 +  s * vvecE1)$
         [ 6.0 ]   [   0.0   ]   [ - 3.0 s ]   [     6.0 - 3.0 s     ]
         [     ]   [         ]   [         ]   [                     ]
         [ 2.0 ] + [  4.0 r  ] + [  4.0 s  ] = [ 4.0 s + 4.0 r + 2.0 ]
         [     ]   [         ]   [         ]   [                     ]
         [ 6.0 ]   [ - 4.0 r ]   [   0.0   ]   [     6.0 - 4.0 r     ]


(%i13) "Drei Punkte A, B, C der Ebene"$

(%i14) A : transpose(matrix ([0.0,0.0,0.0]));
                                    [ 0.0 ]
                                    [     ]
(%o14)                              [ 0.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i15) B : transpose(matrix ([0.0,0.0,6.0]));
                                    [ 0.0 ]
                                    [     ]
(%o15)                              [ 0.0 ]
                                    [     ]
                                    [ 6.0 ]

(%i16) C : transpose(matrix ([6.0,6.0,0.0]));
                                    [ 6.0 ]
                                    [     ]
(%o16)                              [ 6.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i17) "Aufpunkt stuetzvec"$

(%i18) stuetzE2 : transpose(matrix ([0.0,0.0,0.0]));
                                    [ 0.0 ]
                                    [     ]
(%o18)                              [ 0.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i19) "Richtungsvektoren uvec und vvec"$

(%i20) uvecE2 : B - A;
                                    [ 0.0 ]
                                    [     ]
(%o20)                              [ 0.0 ]
                                    [     ]
                                    [ 6.0 ]

(%i21) vvecE2 : C - A;
                                    [ 6.0 ]
                                    [     ]
(%o21)                              [ 6.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i22) "Ebenengleichung"$

(%i23) display(stuetzE2 +  r * uvecE2 +  s * vvecE2)$
                  [ 0.0 ]   [  0.0  ]   [ 6.0 s ]   [ 6.0 s ]
                  [     ]   [       ]   [       ]   [       ]
                  [ 0.0 ] + [  0.0  ] + [ 6.0 s ] = [ 6.0 s ]
                  [     ]   [       ]   [       ]   [       ]
                  [ 0.0 ]   [ 6.0 r ]   [  0.0  ]   [ 6.0 r ]


(%i24) "Schnittgerade der Ebenen E1 und E2"$

(%i25) display(transpose(matrix([0.0,  0.000000000000, 16.000000000000])) +  r * transpose(matrix([-3.0,-3.0,7.0])))$
                   [ 0.0  ]   [ - 3.0 r ]   [   - 3.0 r    ]
                   [      ]   [         ]   [              ]
                   [ 0.0  ] + [ - 3.0 r ] = [   - 3.0 r    ]
                   [      ]   [         ]   [              ]
                   [ 16.0 ]   [  7.0 r  ]   [ 7.0 r + 16.0 ]


(%i26) "Schnittwinkel alpha =  83.034722718821"$

(%i27) "Nr. 39b)"$

(%i28) "Drei Punkte A, B, C der Ebene"$

(%i29) A : transpose(matrix ([0.0,0.0,3.0]));
                                    [ 0.0 ]
                                    [     ]
(%o29)                              [ 0.0 ]
                                    [     ]
                                    [ 3.0 ]

(%i30) B : transpose(matrix ([6.0,0.0,0.0]));
                                    [ 6.0 ]
                                    [     ]
(%o30)                              [ 0.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i31) C : transpose(matrix ([0.0,3.0,0.0]));
                                    [ 0.0 ]
                                    [     ]
(%o31)                              [ 3.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i32) "Aufpunkt stuetzvec"$

(%i33) stuetzE3 : transpose(matrix ([0.0,0.0,3.0]));
                                    [ 0.0 ]
                                    [     ]
(%o33)                              [ 0.0 ]
                                    [     ]
                                    [ 3.0 ]

(%i34) "Richtungsvektoren uvec und vvec"$

(%i35) uvecE3 : B - A;
                                   [  6.0  ]
                                   [       ]
(%o35)                             [  0.0  ]
                                   [       ]
                                   [ - 3.0 ]

(%i36) vvecE3 : C - A;
                                   [  0.0  ]
                                   [       ]
(%o36)                             [  3.0  ]
                                   [       ]
                                   [ - 3.0 ]

(%i37) "Ebenengleichung"$

(%i38) display(stuetzE3 +  r * uvecE3 +  s * vvecE3)$
        [ 0.0 ]   [  6.0 r  ]   [   0.0   ]   [         6.0 r         ]
        [     ]   [         ]   [         ]   [                       ]
        [ 0.0 ] + [   0.0   ] + [  3.0 s  ] = [         3.0 s         ]
        [     ]   [         ]   [         ]   [                       ]
        [ 3.0 ]   [ - 3.0 r ]   [ - 3.0 s ]   [ - 3.0 s - 3.0 r + 3.0 ]


(%i39) "Drei Punkte A, B, C der Ebene"$

(%i40) A : transpose(matrix ([6.0,6.0,0.0]));
                                    [ 6.0 ]
                                    [     ]
(%o40)                              [ 6.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i41) B : transpose(matrix ([6.0,3.0,0.0]));
                                    [ 6.0 ]
                                    [     ]
(%o41)                              [ 3.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i42) C : transpose(matrix ([0.0,6.0,2.0]));
                                    [ 0.0 ]
                                    [     ]
(%o42)                              [ 6.0 ]
                                    [     ]
                                    [ 2.0 ]

(%i43) "Aufpunkt stuetzvec"$

(%i44) stuetzE4 : transpose(matrix ([6.0,6.0,0.0]));
                                    [ 6.0 ]
                                    [     ]
(%o44)                              [ 6.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i45) "Richtungsvektoren uvec und vvec"$

(%i46) uvecE4 : B - A;
                                   [  0.0  ]
                                   [       ]
(%o46)                             [ - 3.0 ]
                                   [       ]
                                   [  0.0  ]

(%i47) vvecE4 : C - A;
                                   [ - 6.0 ]
                                   [       ]
(%o47)                             [  0.0  ]
                                   [       ]
                                   [  2.0  ]

(%i48) "Ebenengleichung"$

(%i49) display(stuetzE4 +  r * uvecE4 +  s * vvecE4)$
             [ 6.0 ]   [   0.0   ]   [ - 6.0 s ]   [ 6.0 - 6.0 s ]
             [     ]   [         ]   [         ]   [             ]
             [ 6.0 ] + [ - 3.0 r ] + [   0.0   ] = [ 6.0 - 3.0 r ]
             [     ]   [         ]   [         ]   [             ]
             [ 0.0 ]   [   0.0   ]   [  2.0 s  ]   [    2.0 s    ]


(%i50) "Schnittgerade der Ebenen E3 und E4"$

(%i51) display(transpose(matrix([0.0,  1.000000000000,  2.000000000000])) +  r * transpose(matrix([-6.0,1.0,2.0])))$
                    [ 0.0 ]   [ - 6.0 r ]   [   - 6.0 r   ]
                    [     ]   [         ]   [             ]
                    [ 1.0 ] + [  1.0 r  ] = [ 1.0 r + 1.0 ]
                    [     ]   [         ]   [             ]
                    [ 2.0 ]   [  2.0 r  ]   [ 2.0 r + 2.0 ]


(%i52) "Schnittwinkel alpha =  42.450155542891"$