Lösungen/Solutions/E1S2AlUe/Ex8

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A  =Anhang/  Annexe / Appendice

Remove["Global`"]

Aufgabe 1

a

Remove[Dz]

α1 = Pi/5; α2 = Pi/6;

1

Dz[α_] := {{Cos[α], -Sin[α], 
          0}, {Sin[α], Cos[α], 0}, {0, 0, 1}};  
    MatrixForm[Dz[α1]]

2

Sy = {{1, 0, 0}, {0, -2, 0}, {0, 0, 1}}; MatrixForm[Sy]

3

MatrixForm[Dz[-α1]]

4

Rollentausch von y- und z-Achse:

R = {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}; MatrixForm[R]

Dy[α_] := R . Dz[α] . Inverse[R];  
    MatrixForm[Dy[α1]]

Dy[α2] . Transpose[{{0, 0, 1}}] // N // 
      MatrixForm

Dy[α2] . Transpose[{{1, 0, 0}}] // N // 
      MatrixForm

Dy[α2] . {1, 0, 0} // N

Resultat

M = Dy[α2] . Dz[-α1] . Sy . Dz[α1]; 
    M // MatrixForm

M // N // MatrixForm

b

Pa = Transpose[{{1, -2, 4}}]; Pb = Transpose[{{5, 5, 8}}]; 
    Pc = Transpose[{{-4, 1, -3}}];

M . Pa // N // MatrixForm

M . Pb // N // MatrixForm

M . Pc // N // MatrixForm

c

vecLen[x_] := Sqrt[x . x]

A1 = vecLen[Cross[Flatten[Pc - Pa], Flatten[Pc - Pb]]]

A1 // N

A2 = vecLen[Cross[Flatten[M . Pc - M . Pa], Flatten[M . Pc - M . Pb]]]; 
    A2 // N

Aufgabe 2

M = {{2, 3}, {4, 5}};  MatrixForm[M]

r[t_] := Transpose[{{-2, 1}} +  t {{5, 7}}]; 
    r[t] // MatrixForm

a

M . r[t] // Expand // MatrixForm

b

MatrixForm[M . {{x}, {y}}] == MatrixForm[{{x}, {y}}]

Solve[M . {{x}, {y}} == {{x}, {y}}, {x, y}]