Lösungen

1

Remove["Global`*"]

Dreh[φ_] := {{Cos[φ], -Sin[φ]}, {Sin[φ], Cos[φ]}};  
    Dreh[φ] // MatrixForm

P1 = {2, 1};  P2 = {3, 2};  P3 = {1, 3};
φ = 71 Degree; 
Q1 = Dreh[φ] . P1 // N

Q2 = Dreh[φ] . P2 // N

Q3 = Dreh[φ] . P3 // N

o = {0, 0};
Show[
        Graphics[
{
Line[{{-1, 
                  0}, {1, 0}}], 
            Line[{{0, -1}, {0, 1}}], 
Line[{P1, P2, P3, 
                P1}], Line[{Q1, Q2, Q3, Q1}], 
Line[{o, 
                P1}], Line[{o, P2}], 
            Line[{o, P3}], 
Line[{o, Q1}], Line[{o, Q2}], 
            Line[{o, Q3}], 
PointSize[0.03], Point[P1], 
            Point[P2], Point[P3], Point[Q1], Point[Q2], 
            Point[Q3]
}
], 
        AspectRatio -> Automatic];

2

α[x_] := 
      ArcTan[x[[2]]/x[[1]]]
x1 = {4, 1}; α[x1]

% // N

S[0] = {{1, 0}, {0, -1}};  S[0] // MatrixForm

S[α_] := 
      Dreh[α] . S[0] . Dreh[-α];   S[α] // 
        Simplify // MatrixForm

S1 = S[α[x1]] . P1

S1 // N

S2 = S[α[x1]] . P2

S2 // N

S3 = S[α[x1]] . P3

S2 // N

Show[
        Graphics[
{
Line[{{-1, 
                  0}, {1, 0}}], 
            Line[{{0, -1}, {0, 1}}], 
Line[{P1, P2, P3, 
                P1}], Line[{S1, S2, S3, S1}], 
Line[{o, 
                x1}], 
Line[{P1, S1}], Line[{P2, S2}], 
            Line[{P3, S3}], 
PointSize[0.03], Point[P1], 
            Point[P2], Point[P3], Point[S1], Point[S2], 
            Point[S3]
}
], 
        AspectRatio -> Automatic];

3

Eigensystem[S[α[x1]]]

\({4, 
          1}\ ist\ x1\ gestreckt\ \((Richtung\ der\ Spiegelungsgerade)\) . \ {\(-\(1\/4\)\), 1}\ \ ist\ darauf\ senkrecht . \ Der\ Eigenwert\  - 
      1\ bewirkt\ die\ Spiegelung, \ 
    der\ Eigenwert\ 1\ das\ Fixhalten\ der\ Abbildung\ senkrecht\ zur\ \(\(Speigelungsachse\)\(.\)\)\)

4

Remove["Global`*"]

vec0 = {0, 0, 0};
a = {3, 1, 2};  b = {-1, 2, -2};  u = {2, 3, -1}; 
    M = Transpose[{a, b, u}] ;  Det[M]

a, b, u sind linear unabhängig, da die Determinante nicht 0 ist.

Proj.Transpose[{a,b,u}]=Transpose[{a,b,0}] ==> Proj = Transpose[{a,b,0}].Inverse[M]

M // MatrixForm

Transpose[{a, b, vec0}] // MatrixForm

vec0 = {0, 0, 0};
 Proj  = Transpose[{a, b, vec0}] . Inverse[M];  
    Proj // MatrixForm

% // N // MatrixForm

T1 = {0, 2, 3};   T2 = {1, 1, 0};  T3 = {2, 0, 2};

N1 = Proj . T1

% // N

N2 = Proj . T2

% // N

N3 = Proj . T3

% // N

o = {0, 0, 0};
Show[
        Graphics3D[
{
Line[{{-1, 0, 
                  0}, {1, 0, 0}}], Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}], 
Line[{N1, 
                N2, N3, N1}], 
            Line[{T1, T2, T3, T1}], 
Line[{o, 
                u}], 
Line[{T1, N1}], Line[{T2, N2}], 
            Line[{T3, N3}], 
PointSize[0.02], Point[T1], 
            Point[T2], Point[T3], PointSize[0.035], Point[N1], Point[N2], 
            Point[N3]
}
], 
        AspectRatio -> Automatic, 
        PlotRange -> {{-1, 9}, {-1, 15}, {-2, 
              4}} ];

5

M1 = (T1 + N1)/2

N[%]

M2 = (T2 + N2)/2

N[%]

M3 = (T3 + N3)/2

N[%]

Matr = 
      Transpose[{M1, M2, M3}] . Inverse[Transpose[{T1, T2, T3}]];   Matr // 
      MatrixForm

N[%] // MatrixForm

Eigensystem[Matr] // N

a={3,1,2}; b={-1,2,-2}; u={2,3,-1};
Auf den Ersten Blick fällt auf, dass unter den Eigenvektoren -u und daher u (gestreckt) vorkommt. Der dazugehörige Eigenwert ist 0.5, während die andern beiden 1 sind (keine Streckung in diese Richtungen).

Matr . T1

Richtig

Matr . T2

Richtig

Matr . T3

Richtig

6

Remove["Global`*"]

a = {3, 1, 2};
b1 = a/Norm[a]

{1, y1, 0} . a == 0

Solve[{1, y1, 0} . a == 0, {y1}]

b2work = {1, -3, 0}

b2 = b2work/Norm[b2work]

b3 = Cross[b1, b2]

U = Transpose[{b1, b2, b3}];  U // MatrixForm

Dreh0[φ_] := {{1, 0, 0}, {0, 
          Cos[φ], -Sin[φ]}, {0, Sin[φ], 
          Cos[φ]}};  Dreh0[φ] // MatrixForm

DrehAchse[φ_] := 
      U . Dreh0[φ] . Inverse[U]

Dreh = DrehAchse[56 Degree] // N;  Dreh // MatrixForm

T1 = {0, 2, 3};   T2 = {1, 1, 0};  T3 = {2, 0, 2};

R1 = Dreh . T1

R2 = Dreh . T2

R3 = Dreh . T3

o = {0, 0, 0};
Show[
        Graphics3D[
{
Line[{{-1, 0, 
                  0}, {1, 0, 0}}], Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 
                  1}}], 
Line[{-4 b1, 4  b1}], 
            Line[{-2 b2, 2  b2}], 
            Line[{-2 b3, 2  b3}], 
Line[{R1, R2, R3, 
                R1}], Line[{T1, T2, T3, T1}], 
PointSize[
              0.02], Point[T1], Point[T2], Point[T3], PointSize[0.035], 
            Point[R1], Point[R2], 
            Point[R3]
}
], 
        AspectRatio -> Automatic ];

7

a  Berechnung der Punkte

Remove["Global`*"]

a = {3, 1, 2};
b = {-1, 3, 0};
c = Cross[a, b]

B = Transpose[{a, b, c}];   B // MatrixForm

Dλ = {{1, 0, 0}, {0, 1, 0}, {0, 0, -1}};  
    Dλ // MatrixForm

A = B . Dλ . Inverse[B];  A // MatrixForm

T1 = {0, 2, 3};   T2 = {1, 1, 0};  T3 = {2, 0, 2};

S1 = A . T1

S2 = A . T2

S3 = A . T3

M1 = (T1 + S1)/2

M2 = (T2 + S2)/2

M3 = (T3 + S3)/2

o = {0, 0, 0};
Show[
        Graphics3D[
{
Line[{{-1, 0, 
                  0}, {1, 0, 0}}], Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}], 
Line[{S1, 
                S2, S3, S1}], Line[{T1, T2, T3, T1}], 
            Line[{M1, M2, M3, M1}], 
Line[{S1, T1}], 
            Line[{S2, T2}], 
            Line[{S3, T3}], 
PointSize[0.03], Point[T1], 
            Point[T2], Point[T3], PointSize[0.05], Point[S1], Point[S2], 
            Point[S3], PointSize[0.07], Point[M1], Point[M2], Point[M3], 
            PointSize[0.09], 
            Point[o]
}
], 
        AspectRatio -> Automatic, 
        ViewPoint -> {0.478,  4.341,  0.259} ];

b  Kontrolle: Liegen die Punkte (Subscript[M, k])in der Ebene Φ?

Solve[λ a  +  μ b  +  ν c  ==  
        M1, {λ, μ, ν}]

Solve[λ a  +  μ b  +  ν c  ==  
        M2, {λ, μ, ν}]

Solve[λ a  +  μ b  +  ν c  ==  
        M3, {λ, μ, ν}]

In Ordnung, denn der Koeffizient ν für die Richtung c ist immer 0.