Lösungen/Solutions/
F1S1Ue/Ex14

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A  =Anhang/  Annexe / Appendice

Remove["Global`@*"]

Ueb/Ex 1      W

Φ1 = (x + y + z + 1 == 
            0);
Φ2 = (x + y + 2  z - 2 == 
            0);
Φ3 = (x + 2  y + 4  z + 10 == 
            0);
Φ4 = (-x + y - 3  z + 7 == 
            0);

solvP1 = 
        Solve[{Φ2, Φ3, Φ4}, {x, y, z}] // 
          Flatten;
P1 = {x, y, z} /. solvP1

N[%]

solvP2 = 
        Solve[{Φ1, Φ3, Φ4}, {x, y, z}] // 
          Flatten;
P2 = {x, y, z} /. solvP2

N[%]

solvP3 = 
        Solve[{Φ1, Φ2, Φ4}, {x, y, z}] // 
          Flatten;
P3 = {x, y, z} /. solvP3

solvP4 = 
        Solve[{Φ1, Φ2, Φ3}, {x, y, z}] // 
          Flatten;
P4 = {x, y, z} /. solvP4

vecLen[r_] := Sqrt[r . r]

n = {1, 1, 1};  r = {x, y, z};
h1[r_] :=  (r . n + 1)/vecLen[n];   h1[r]

d = h1[{0, 0, 0}]

N[%]

h = h1[P1]

N[%]

Ueb/Ex 2      W

z[t_] :=  5 E^(I (Pi/3 t  +  Pi/5))

ParametricPlot[
        Evaluate[{Re[z[t]], Im[z[t]]}], {t, 0, 
          Pi}, 
AspectRatio -> 1];

ParametricPlot[
        Evaluate[{Re[z[t]], Im[z[t]]}], {t, 0, 
          2  Pi}, 
AspectRatio -> 1];

Ueb/Ex 3      W

Remove[w, p]

w[n_] := ( E^(I 2 n Pi/5))

w[2]

Im[w[2]]

Show[
        Graphics[
          Join[{PointSize[0.03], Line[{{0, -1}, {0, 1}}], 
              Line[{{-1, 0}, {1, 0}}], Circle[{0, 0}, 1]}, 
            Map[Point, Table[p[n], {n, 0, 4}]]]], 
        AspectRatio -> 1];

Ueb/Ex 4      W

Remove[w, p]

w[n_] := (5^(1/
                5)  E^(I ( Pi/6 + 2 n Pi)/5))

w[2]

Re[w[2]]

Im[w[2]]

p[n_] := {Re[w[n]], Im[w[n]]};  p[1]

Show[
        Graphics[
          Join[{PointSize[0.03], Line[{{0, -1.5}, {0, 5}}], 
              Line[{{-1.5, 0}, {5, 0}}], 
              Circle[{0, 0}, 
                5^(1/5)], 
Point[{Re[
                    5 E^(I Pi/6)], 
                  Im[5 E^(I Pi/6)]}], 
Line[{{0, 
                    0}, {Re[5 E^(I Pi/6)], Im[5 E^(I Pi/6)]}}]}, 
            Map[Point, Table[p[n], {n, 0, 4}]]]], 
        AspectRatio -> 01];

Ueb/Ex 5      W

a

{z1 = 5 E^(I Pi/6), z2 = 1/3 E^(I Pi 4/5)}

z1 * z2

b

z1/z2

Show[
        Graphics[{PointSize[0.03], Line[{{0, -1.5}, {0, 1.5}}], 
            Line[{{-1.5, 0}, {1.5, 0}}], 
            Circle[{0, 0}, 1], 
Point[{Re[z1], Im[z1]}], 
            Point[{Re[z2], Im[z2]}], 
Point[{Re[z1 z2], 
                Im[z1 z2]}], 
            Point[{Re[z1/z2], 
                Im[z1/z2]}]
}
], 
        AspectRatio -> 1.5];

z1^2 * z2^3

Re[z1^2 / z2^3]

N[%]

Im[z1^2 / z2^3]

N[%]