Lösungen/Solutions

Remove["Global`*"]

Aufgabe 1

Flatten[Solve[z1 z  +  1/z2 == Conjugate[z1]/Re[z1], {z}], 
      2]

z1 = 2 + 4  I;  z2 = -4 - 6  I;

a

Conjugate[z1]

1/z2

N[%]

Re[z1]

zRes = 
      z /. Flatten[Solve[z1 z  +  1/z2 == Conjugate[z1]/Re[z1], {z}], 
          2];  zRes

N[%]

b

p[z_] := Point[{Re[z], Im[z]}]

Show[
        Graphics[
{PointSize[0.03], 
p[
              z1], p[z2], p[z2^(-1)], p[Conjugate[z1]], p[Re[z1]], 
            p[zRes], 
Line[{{0, -7}, {0, 7}}], 
            Line[{{-7, 0}, {7, 0}}], 
Circle[{0, 0}, 
              1]
}], AspectRatio -> 01];

Show[
        Graphics[
{PointSize[0.03], 
p[
              z2^(-1)], 
            p[zRes], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[{0, 
                0}, 1]
}], 
        AspectRatio -> 01];

Solve[w^6 == z2, {w}]

solv1 = Solve[w^6 == z2, {w}] // N // Flatten

w[n_] := (w /. solv1[[n]])

w[4]

Show[
        Graphics[
Join[{PointSize[
                0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
              Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[{0, 
                  0}, 1]
}, Table[p[w[n]], {n, 6}]]], 
        AspectRatio -> 01];

Show[
        Graphics[
Join[{PointSize[0.03], 
              p[z2], 
Line[{{0, -7}, {0, 7}}], 
              Line[{{-7, 0}, {7, 0}}], 
Circle[{0, 0}, 
                1], p[z2]
}, Table[p[w[n]], {n, 6}]]], 
        AspectRatio -> 1];

Aufgabe 2

Remove["Global`*"]

rot[{x_, y_}, φ_] := {Cos[φ] x  -  
            Sin[φ] y,  
          Cos[φ] y +  
            Sin[φ] x};
φ = 35/360  *2 Pi

p[z_] := Point[{z[[1]], z[[2]]}]

a

vecLen[a_] := Sqrt[a . a]

vecA = {4, 3}; vecB = {5, 7}; vecC = {1, 6};

vecAC = vecC - vecA; vecBC = vecC - vecB; 
    vecAB = vecB - vecA

vecCA = -vecAC; vecCB = -vecBC; vecBA = -vecAB;

nAB = {-4, 1}

rA = rot[vecA, φ]

vA = N[%]

rB = rot[vecB, φ]

vB = N[%]

rC = rot[vecC, φ]

vC = N[%]

{vA, vB, vC}

Show[
        Graphics[
{PointSize[0.03], 
p[
              vecA], p[vecB], p[vecC], p[rA], p[rB], 
            p[rC], 
Line[{{0, -9}, {0, 9}}], 
            Line[{{-9, 0}, {9, 0}}], 
Line[{vecA, 
                vecB}], Line[{vecA, vecC}], 
            Line[{vecC, vecB}], 
Line[{rA, rB}], 
            Line[{rA, rC}], 
            Line[{rC, rB}], 
Line[{{0, 0}, vecA}], 
            Line[{{0, 0}, rA}]
}], 
        AspectRatio -> 01];

b

Solve[vecA . nAB == vecLen[vecAB]*d, {d}] // Flatten

N[%]

c

grad[φ_] := N[φ /(2  Pi) 360]

α = 
      ArcCos[vecAC . vecAB/(vecLen[vecAC] vecLen[vecAB])]

N[%]

grad[α]

β = 
      ArcCos[vecBC . vecBA/(vecLen[vecBC] vecLen[vecBA])]

N[%]

grad[β]

γ = 
      ArcCos[vecCA . vecCB/(vecLen[vecCA] vecLen[vecCB])]

N[%]

grad[γ]

grad[α] + grad[β] + grad[γ]

d

A = Det[{vecAB, vecAC}]/2

A = Det[{vecBC, vecBA}]/2

A = Det[{vecCA, vecCB}]/2

% // N

Aufgabe 3

a

r[x_, y_, z_] = {x, y, z};

nΦ = {-2, 4, -3};  
    nΨ = {-1, -2, 4};

Φ[x_, y_, z_] := 
        r[x, y, z] . nΦ + 2;
Ψ[x_, y_, z_] := 
        r[x, y, z] . nΨ - 1;

Sxy = 
      Solve[{Φ[x, y, 0] == 
              0, Ψ[x, y, 0] == 0}, {x, y}]  // 
        Flatten

P0 = {x, y, 0} /.  Sxy

N[P0]

n = Cross[nΦ,  nΨ]

s[t_] := P0 + t n;  s[t]

b

vecLen[a_] := Sqrt[a . a]

Φ[x, y, z]

hΦ[{x_, y_, z_}] := (Φ[x, y, z])/
        vecLen[nΦ];  hΦ[{x, y, z}]

Pp = {4, 4, 4}

distP0Φ = hΦ[Pp]

N[%]

c

Remove[vecP0s]

Pp = {4, 4, 4}

vecP0s[t_] := s[t] - Pp

solv2b = Solve[vecP0s[t] . n == 0, {t}] // Flatten

distP0s = vecP0s[t] /. solv2b

N[distP0s]

vecLen[distP0s]

N[%]

Andere, kürzere Lösung:

distP0s = vecLen[Cross[n, Pp - P0]]/vecLen[n]

N[%]