Lösungen/Solutions/
F1S1Ue/Ex11

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A  =Anhang/  Annexe / Appendice

Remove["Global`@*"]

Ueb/Ex 1      W

syst = {
a ==    3 e1  -  2 e2  +  
              5 e3, 
b ==    7 e1  +     e2  -     e3, 
c ==  -4 e1  +  3 e2  -  
              6 e3};

solv = Solve[syst, {e1, e2, e3}] // Flatten

d = 1 e1  +  2 e2  +  5 e3  /.  solv  // Simplify

dBap = 1/12 {107, 1, 79}

Ueb/Ex 2      W

vec = {OA, OB, OC}

AB = OB - OA; BA = -AB;  BC = OC - OB; CB = -BC;  
    CA = OA - OC; AC = -CA;

{AB, BA, BC, CB, CA, AC}

OMa = OB + 1/2 BC; OMb = OC + 1/2 CA; 
    OMc = OA + 1/2 AB;
MaA = OA - OMa;  MbB = OB - OMb;  McC = OC - OMc;

{OMa, OMb, OMc, MaA, MbB, McC} // Simplify

gc[λ_] := 
      OMc +  λ McC;  gb[μ_] := 
      OMb +  μ MbB;

proOS =  (gc[λ] ==  gb[μ]  // Simplify)

Wähle KS: Ursprung in A und OB nicht parallel OC ⇒  OA = Nullvektor =0
• Choisir syst. de coord. : Origine à A et OB non parallel à OC ⇒  OA = vecteur zéro=0

OA = 0; proOS

OB nicht parallel OC ⇒  OB, OC linear unabhängig ⇒ Koeff . in Gleichung = 0
• OB non parallel à OC ⇒  OB, OOC linéairement independant ⇒ Coeff. dans l'équation = 0

Solve[{-1 + 2 λ + μ == 
          0,  -1 + λ + 2 μ == 
          0}, {λ, μ}]

Clear[OA]

OS = gc[(1/3)] // Simplify

OS = gb[(1/3)] // Simplify

Ueb/Ex 3      W

(Sin[α] + Cos[α])^2

exprExp2 = (Sin[α] + Cos[α])^2  // 
        TrigReduce

maxExprExp2 =  2

maxExpr = Sqrt[maxExprExp2]

Ueb/Ex 4      W

Clear[φ]

a = Transpose[{{2, 3}}];  
    rot = {{Cos[φ], -Sin[φ]}, {Sin[φ], 
          Cos[φ]}}

aNew = rot . a

Bsp. • Exp.

φ =  Pi/6

aNew = rot . a //  Simplify  // Flatten

% // N

Show[
        Graphics[{PointSize[0.03], Line[{{0, -4}, {0, 4}}], 
            Line[{{-4, 0}, {4, 0}}], 
            Line[{{0, 0}, {2, 3}}], 
Line[{{0, 
                  0}, {-(3/2) + (Sqrt[3]), 
                  1 + ((3 (Sqrt[3]))/2)}}], 
Point[{2, 3}], 
            Point[{-(3/2) + (Sqrt[3]), 1 + ((3 (Sqrt[3]))/2)}]}], 
        AspectRatio -> 1];

Ueb/Ex 5      W

γ = π - β - α

Tan[α] + Tan[β] + Tan[γ] == 
        Tan[α] Tan[β] Tan[γ] //  TrigReduce

tan[
        x_ + y_] := (tan[x] + tan[y])/(1 - tan[x] tan[y])

Lexpr = 
      tan[α] + tan[β] - tan[α + β]

Lexpr1 = Simplify[Lexpr]

Lexpr2 = Expand[Lexpr1] // Together // Factor

Lexpr3 = Lexpr2 /.  tan[x_] -> tanNew[x]

Lexpr4 = 
      Lexpr3 /. (tanNew[x_] + tanNew[y_])/(-1 + 
                tanNew[x_] tanNew[y_]) -> -tanNew[x + y]

Ueb/Ex 6      W

Remove[t]

v[t_] := {3, 2} +  t {4, -1}

tSolv = 
      Solve[v[t][[1]] == {x, y}[[1]], {t}]  // 
        Flatten

v[t][[2]] == {x, y}[[2]]  /.  tSolv  // 
      Simplify

1 == x/11  +  4  y/11

x0 = 1/(1/11)

y0 = 1/(4/11)

% // N

Plot[11/4 - x/4, {x, -1, 12}];

ParametricPlot[Evaluate[v[t]], {t, -1.3, 2.3}];