Lösungen

Remove["Global`*"]

Aufgabe 1

a

r[x_, y_, z_] := {x, y, z};
x[λ_, μ_] := 
        3 + 4 λ - 2 μ;
y[λ_, μ_] := α + λ + μ;
z[λ_, μ_] := -2 + μ;
r[x[λ, μ], y[λ, μ], 
      z[λ, μ]]

solv1a = 
      Solve[r[x[λ, μ], y[λ, μ], 
              z[λ, μ]] == {4, 6, 
              8}, {α, λ, μ}] // Flatten

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b

x1[ν_, λ_, μ_] := 
        3  ν + 4 λ - 2 μ;
y1[ν_, λ_, μ_] := α ν + λ + μ;
z1[ν_, λ_, μ_] := -2 ν + μ;

Solve[
        r[x1[ν, λ, μ], y1[ν, λ, μ], 
            z1[ν, λ, μ]] == {4, 6, 
            8}, {ν, λ, μ}] // Flatten

Solve[9 + 4 α == 0, {α}]  //  Flatten

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Aufgabe 2

a

Remove["Global`*"]

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

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

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]

c

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

hΦ[{x_, y_, z_}] := (r[x, y, z] . nΦ - 2)/
        vecLen[nΦ];  hΦ[x, y, z]

Pp = {4, 4, 4}

distP0Φ = hΦ[Pp]

N[%]

b

Remove[vecP0s]

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]

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Aufgabe 3

rot[{x_, y_}, φ_] := {Cos[φ] x  -  
            Sin[φ] y,  
          Cos[φ] y +  
            Sin[φ] x};
φ = 1.8157749899217608

Show[
        Graphics[
{PointSize[
              0.03], 
Point[p1 = {4, 3}], 
            Point[p2 = {5, 7}], 
            Point[p3 = {1, 6}], 
Point[
              r1 = rot[p1, φ]], Point[r2 = rot[p2, φ]], 
            Point[r3 = rot[p3, φ]], 
Line[{p1, 
                p2}], Line[{p3, p2}], 
            Line[{p1, p3}], 
Line[{r1, r2}], 
            Line[{r1, r3}], 
            Line[{r3, r2}], 
Line[{{0, -9}, {0, 9}}], 
            Line[{{-9, 0}, {9, 0}}], 
Line[{p1, 
                p1 - {-(p2 - p1)[[2]], (p2 - 
                          p1)[[1]]}}], 
Line[{r1, 
                r1 - {-(r2 - r1)[[2]], (r2 - 
                          r1)[[1]]}}], 
Line[{p1, {0, 
                  0}}], Line[{r1, {0, 0}}], 
Circle[{0, 0}, 
              Sqrt[5^2 + 7^2]]
}], 
        AspectRatio -> 01];

a

Remove["Global`*"]

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

AB = {5 - 4, 7 - 3}

nAB = {-4, 1}

eyAxe = {0, -1}

φ = 
      ArcCos[nAB . eyAxe/(vecLen[nAB] vecLen[eyAxe])]

N[%]

% 360/(2  Pi)

b

ANew = {4 Cos[φ] - 3 Sin[φ], 
        4 Sin[φ] + 3  Cos[φ]}

N[%]

c

vecA = {4, 3, 0}; vecB = {5, 7, 0}; vecC = {1, 6, 0}; 
    vecD[z_] := {6, 5, z};

Vol[z_] := Det[{vecB - vecA, vecC - vecA, vecD[z] - vecA}]/3 /
        2

Vol[z]

Solve[Vol[z] == 25, {z}] //  Flatten

N[%]