Lösungen/Solutions/
F1S1Ue/Ex13

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

Remove["Global`*"]

Ueb/Ex 1      W

(Problem schon in Serie 12 / Problème déjà dans la série 12)

rg1[t_] := {-2, 4} + t {7, 2};
rg2[t_] := {8, 3} + t {-4, 5};

s1a = 
      Solve[rg1[λ] == rg2[μ], {λ, μ}] // 
        Flatten

s1a // N

res1a = {rg1[λ], rg2[μ]} /. s1a

res1a // N

ParametricPlot[Evaluate[rg1[t]], {t, -4, 4}];

ParametricPlot[
        Evaluate[{rg1[t], rg2[t]}], {t, -4, 4}];

h[t_] := 
      rg1[t] + {-2, 7}/Sqrt[(-2)^2  +  7^2] 4

h[t] // N

ParametricPlot[
        Evaluate[{rg1[t], rg2[t], h[t]}], {t, -4, 4}];

s1b = 
      Solve[h[λ] == rg2[μ], {λ, μ}] // 
        Flatten

s1b // N

{h[λ], rg2[μ]} /. s1b  // N

φ =  
      ArcCos[{7, 2} . {-4, 5}/
          Sqrt[({7, 2} . {7, 2}) ({-4, 5} . {-4, 
                    5})]]

N[φ]

N[φ]/Degree

Ueb/Ex 2      W

a

g[t_] := {-2, 4, 1} + t {7, 2, -1}

Φ[λ_, μ_] := {1, 
          0, -2} + λ {4, 1, 4} + μ {-3, -2, 
            1}

sSolv = 
      Solve[Φ[λ, μ] == 
            g[t], {t, λ, μ}] //  Flatten

sSolv[[1]]

S = g[t]  /.  sSolv[[1]]

b

Remove[s]

Ψ[λ_, μ_] := {0, 1, 
          1} + λ {-1, 2, 3} + μ {1, -1, 1}

Φ[λ_, μ_] := {1, 
          0, -2} + λ {4, 1, 4} + μ {-3, -2, 
            1}

sSolv = Solve[Ψ[λ, μ] == Φ[λ1, μ1], {λ1, μ1, μ}]//Flatten

s[λ_] := Φ[λ, 
          (2/5) (5 + 7 λ)]  //  Simplify;  s[t]

ParametricPlot3D[s[t], {t, -3, 3}];

{x, y, 0} == s[t]

h1Solv = 
      Solve[{x, y, 0} == s[t], {t, x, y}]  // Flatten

s[t] /.  h1Solv

h2Solv = 
      Solve[{0, y, z} == s[t], {t, y, z}]  // Flatten
{t -> 0, x -> -5, y -> -4}

s[t] /.  h2Solv

N[%]

h3Solv = 
      Solve[{x, 0, z} == s[t], {t, x, z}]  // Flatten
{t -> 0, x -> -5, y -> -4}

s[t] /.  h3Solv

N[%]

Ueb/Ex 3      W

Remove[α]

φ = Pi/7

a1ProjRot = {1 Cos[φ] -  2 Sin[φ],  
        1 Sin[φ] +  2 Cos[φ], 0}

% // N

aProjRot = {1 Cos[φ] -  2 Sin[φ],  
        1 Sin[φ] +  2 Cos[φ], 3}

% // N

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

ϑ[n_] := {ArcCos[a[[n]]/len[a]],  
        N[ArcCos[a[[n]]/len[a]]]}

a = {1, 2, 3}

α = ϑ[1]

β = ϑ[2]

γ = ϑ[3]

Ueb/Ex 4      W

Remove[Ψ, x, y, z]

Ψ[λ_, μ_] := {0, 1, 
          1} + λ {-1, 2, 3} + μ {1, -1, 1}

Ψ[λ, μ] == {x, y, z}

solv4 =  
      Solve[Ψ[λ, μ] == {x, y, 
              z}, {λ, μ, x}] // Flatten

x1 - x == 0/.solv4[[3]]

Remove[x]

Expand[x + (1/5) (-3 + 4 y - z)] == 0

Expand[5 (x + (1/5) (-3 + 4 y - z))] == 
      0

HNF[{x_, y_, z_}] := (-3 + 5 x + 4 y - z) /
          Sqrt[5^2 + 4^2 + (-1)^2];
HNF[{x, y, z}]

OP0 = {15, 20, 18}

d = HNF[OP0]

n = {5, 4, -1}/Sqrt[42]

gn[t_] := OP0 +  t n;  gn[t]

solv41 =  
      Solve[Ψ[λ, μ] == 
            gn[t], {λ, μ, t}] // Flatten

solv41[[3]]

H = gn[t] /. solv41[[3]]