(%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