Lösungen/ Solutions

Remove["Global`*"]

Aufgabe 1 /Problème 1

Remove["Global`*"]

solv = 
      Solve[{a == e1 + e2 + e3, b == e1 - e2, 
            c == e1 + 2  e2 + 4  e3}, {e1, e2, e3}] // 
        Flatten

% // ExpandAll

% // N

Aufgabe 2 /Problème 2

Remove["Global`*"]

len[v_] := Sqrt[v . v];

a = {0, 0}; b = {8, 0}; c = {5, 6};
ac = c - a; bc = c - b; abb = 2/3 ac;  cbb = ac - abb;  
    baa = 2/5 bc;  caa = bc - baa;
{λ = len[baa]/len[caa], μ = len[ cbb]/len[abb], ν = 
        1/(λ μ)}

s = Solve[ν == x/(8 - x), {x}] // Flatten

cc = {x, 0} /. s

Show[
        Graphics[{Line[{a, b}], Line[{b, c}], Line[{c, a}], Line[{b, abb}], 
            Line[{a, b + baa}], Line[{c, cc}]}]];

Aufgabe 3 /Problème 3

Remove["Global`*"]

a = 67.54;  
    b = 59.18;  γ = (98 + 12/60 + 14/(60^2)) 2 Pi/
          360

N[%]

(98 + 12/60 + 14/(60^2) ) Degree // N

c = Sqrt[a^2 + b^2 - 2 a b Cos[γ]]

α = ArcSin[a Sin[γ]/c]

α/Degree

Aufgabe 4 /Problème 4

Remove["Global`*"]

a

a = t;  b = t; c = q t;

oo = 6; aa = 10 - 6; bb = 16 - 10; 
    cc = 25 - 16;
v1 = -aa/bb;  v2 = (aa + bb + cc)/(cc);

{dv = v1/v2, v1, v2}

{dv, v3 = -a/b, v4 = (a + b + c)/c, v3/v4} // 
      Simplify

s = Solve[v3/v4 == dv, {q}] // Flatten

c/b  /. s

b

b1 = a1; c1 = a1;
dv2 = (-a1/b1)/((a1 + b1 + c1)/(c1))

dv == dv2

Aufgabe 5 /Problème 5

a

Remove["Global`*"]

a = {1, 1};  b = {10, 4};  c = {5, 9};
ac = c - a;  cb = b - c;  ba = a - b;

len[v_] := Sqrt[v . v];

s[u_, v_] := 
      u + 1/2 v +  {-v[[2]], v[[1]]} * 
          1/3 * Sqrt[3]/2

sb = s[a, ac]

N[%]

sa = s[c, cb]

N[%]

sc = s[b, ba]

N[%]

b

len[sa - sb] // Simplify

N[%]

len[sb - sc] // Simplify

N[%]

len[sc - sa] // Simplify

N[%]