Lösungen/ Solutions

Aufgabe 1 /Problème 1

Remove["Global`*"]

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

a[1] = {7, 4}; a[2] = {7, 4}; a[3] = {-4, 10}; 
    a[4] = {2, 4};

q[1] = {0, 4}; q[2] = {0, y2}; q[3] = {4, 0}; q[4] = {2, 0};

ort[a_] := {-a[[2]], a[[1]]}

gc[n_, {x_, y_}] := ort[a[n]] . {x, y} + c[n]

solv[n_] := Flatten[Solve[gc[n, q[n]] == 0, {c[n]}]]

solv[1]

g[n_, {x_, y_}] := (gc[n, {x, y}] /. solv[n])

Table[g[n, {x, y}], {n, 4}]

1a

n1 = ort[a[1]]/vecLen[a[1]]

N[n1]

1b

HNF[n_, {x_, y_}] := g[n, {x, y}]/vecLen[a[n]]

HNF[1, {x, y}]

d1 = 28/Sqrt[65]

N[d1]

1c

HNF[2, {x_, y_}] := HNF[1, {x, y}] - 4;  HNF[2, {x, y}]

solvy2 = Solve[HNF[2, {0, y2}] == 0, {y2}] // Flatten

y2 = y2 /. solvy2

N[y2]

1d

solvOP1 = 
        Solve[{HNF[2, {x, y}] == 0, 
              HNF[3, {x, y}] == 0}, 
{x, y}] // 
          Flatten; 
OP1 = {x, y} /. solvOP1

N[OP1]

1e

solvOP2 = 
        Solve[{HNF[4, {x, y}] == 0, 
              HNF[3, {x, y}] == 0}, 
{x, y}] // 
          Flatten;
OP2 = {x, y} /. solvOP2

N[OP2]

solvOP3 = 
        Solve[{HNF[2, {x, y}] == 0, 
              HNF[4, {x, y}] == 0}, 
{x, y}] // 
          Flatten;
OP3 = {x, y} /. solvOP3

N[OP3]

1f

A[a_, b_] := 
      vecLen[a] vecLen[
          b] *Sin[ArcCos[a . b/(vecLen[a] vecLen[b])]]/
          2

A[OP2 - OP1, OP3 - OP1] // Short

A[OP2 - OP1, OP3 - OP1] // N

Aufgabe 2 /Problème 2

Remove["Global`*"]

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

n2o = {A, B, C}

Ox0 = {3, 0, 0}; Oy0 = {0, 2, 0}; Oz0 = {0, 0, 4};

Φ[1, {x_, y_, z_}] := n2o . {x, y, z} + 1; Φ[
      1, {x, y, z}]

solv[k_, p_] := 
      Flatten[Solve[Φ[1, p] == 0, {k}]]

solv2 = Flatten[{solv[A, Ox0], solv[B, Oy0], solv[C, Oz0]}]

n2o = n2o /. solv2

Φ[1, {x_, y_, z_}] := 
      n2o . {x, y, z} + 1 /. solv2; Φ[1, {x, y, z}]

Φ[2, {x_, y_, z_}] := n2o . {x, y, z} + k; Φ[
      2, {x, y, z}]

solv2a = 
      Flatten[Solve[Φ[2, {5, 0, 0}] == 0, {k}]]

Φ[2, {x_, y_, z_}] := n2o . {x, y, z} + 5/3; Φ[
      2, {x, y, z}]

2a

solv2y2 = 
        Flatten[Solve[Φ[2, {0, y2, 0}] == 
              0, {y2}]];
y2 = y2 /. solv2y2

N[%]

solv2z2 = 
        Flatten[Solve[Φ[2, {0, 0, z2}] == 
              0, {z2}]];
z2 = z2 /. solv2z2

N[%]

2b

n2 = n2o/vecLen[n2o]

% // N

vecLen[n2]

2c

α = ArcCos[n2o[[1]]/vecLen[n2o]]

% // N

2.108479579888103 * 360 /(2 Pi)

β = ArcCos[n2o[[2]]/vecLen[n2o]]

% // N

2.44685437739309 * 360 /(2 Pi)

γ = ArcCos[n2o[[3]]/vecLen[n2o]]

% // N

1.965040701859682 * 360 /(2 Pi)

Cos[α]^2 + Cos[β]^2 + Cos[γ]^2  // N

2d

d1 = 1/vecLen[n2o]

N[%]

d2 = 5/3 /vecLen[n2o]

N[%]

2e

V = 5 y2 z2 /6

N[%]

Remove["Global`*"]

Aufgabe 3 /Problème 3

3a

eqa=(-1 == (-1/6) *(-5/
                4) * (-(11 - q))/
              q )

solv3a = Solve[eqa, {q}] // Flatten;
q = q /.  solv3a

N[%]

p = 11 - q

N[%]

3b

Remove[s]

eqb = (1 == 
          s/(s + 7) *(-5/4)*(-p/q) )

solv3b = Solve[eqb, {s}] // Flatten;
s = s /.  solv3b

N[%]