(%i1) Cornelsen S. 114, Nr. 5

(%i2) Vektor a

(%i3) a: transpose(matrix ([4.0,6.0,2.0]));
                                    [ 4.0 ]
                                    [     ]
(%o3)                               [ 6.0 ]
                                    [     ]
                                    [ 2.0 ]

(%i4) Vektor b

(%i5) b: transpose(matrix ([1.0,4.0,2.0]));
                                    [ 1.0 ]
                                    [     ]
(%o5)                               [ 4.0 ]
                                    [     ]
                                    [ 2.0 ]

(%i6) Kreuzprodukt zwischen den Vektoren: a x b 

(%i7) a_b : transpose(matrix ([4.0,-6.0,10.0]));
                                   [  4.0  ]
                                   [       ]
(%o7)                              [ - 6.0 ]
                                   [       ]
                                   [ 10.0  ]

(%i8) Cornelsen S. 114, Nr. 6

(%i9) Vektor v_AB

(%i10) v_AB: transpose(matrix ([5.0,4.0,1.0])) - transpose(matrix ([3.0,0.0,0.0]));
                                    [ 2.0 ]
                                    [     ]
(%o10)                              [ 4.0 ]
                                    [     ]
                                    [ 1.0 ]

(%i11) Vektor v_AC

(%i12) v_AC: transpose(matrix ([0.0,6.0,3.0])) - transpose(matrix ([3.0,0.0,0.0]));
                                   [ - 3.0 ]
                                   [       ]
(%o12)                             [  6.0  ]
                                   [       ]
                                   [  3.0  ]

(%i13) Vektor v_BC

(%i14) v_BC: transpose(matrix ([0.0,6.0,3.0])) - transpose(matrix ([5.0,4.0,1.0]));
                                   [ - 5.0 ]
                                   [       ]
(%o14)                             [  2.0  ]
                                   [       ]
                                   [  2.0  ]

(%i15) Winkel zwischen Vektor v_AB und Vektor v_AC: alpha =  51.419794203452° 

(%i16) Winkel zwischen Vektor v_AB und Vektor v_BC: alpha =  90.000000000000°

(%i17) Winkel zwischen Vektor v_AC und Vektor v_BC: alpha =  38.580205796548°

(%i18) Also ist das Dreieck rechtwinklig!

(%i19) Cornelsen S. 114, Nr. 7

(%i20) a)

(%i21) x-Vektor xvec

(%i22) xvec : transpose(matrix ([x,y,z]));
                                     [ x ]
                                     [   ]
(%o22)                               [ y ]
                                     [   ]
                                     [ z ]

(%i23) Aufpunkt stuetzvec

(%i24) stuetzg_u : transpose(matrix ([0.0,0.0,0.0]));
                                    [ 0.0 ]
                                    [     ]
(%o24)                              [ 0.0 ]
                                    [     ]
                                    [ 0.0 ]

(%i25) Richtungsvektor uvec

(%i26) uvecg_u : transpose(matrix ([3.0,4.0,3.0]));
                                    [ 3.0 ]
                                    [     ]
(%o26)                              [ 4.0 ]
                                    [     ]
                                    [ 3.0 ]

(%i27) Geradengleichung g

(%i28) display(stuetzg_u +  r * uvecg_u)$
                        [ 0.0 ]   [ 3.0 r ]   [ 3.0 r ]
                        [     ]   [       ]   [       ]
                        [ 0.0 ] + [ 4.0 r ] = [ 4.0 r ]
                        [     ]   [       ]   [       ]
                        [ 0.0 ]   [ 3.0 r ]   [ 3.0 r ]

(%i59) Schnittwinkel g_u und g_x: alpha =  59.036°

(%i62) Schnittwinkel g_u und g_y: alpha =  46.686°

(%i65) Schnittwinkel g_u und g_z: alpha =  59.036°

(%i66) Viel einfacher über die Richtungsvektoren . . .

(%i67) b)

(%i68) cos 45 Grad = 1 / sqrt(2) = 0,7071

(%i69) =>  0,7071 = t / (1 * sqrt(9 + 16 + t^2)

(%i70) =>  0,7071^2 * (9 + 16 + t^2) = t^2

(%i71) =>  0,5 * (9 + 16) = t^2 - 0.5t^2 

(%i72) =>  25 = t^2 

(%i73) =>  t = 5