Lösungen/Solutions/
B1S1Ue/Ex14

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

Remove["Global`*"]

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

Ueb/Ex 1      W

r0 = {0, 3}; a = {3, 1}; aOrth = {-1, 3};  r = {x, y};

n = aOrth;

en = n/vecLen[n]

r . aOrth

h1[{x_, y_}] := - x + 3 y  +  C1 ;   solv1 = 
      Solve[h1[r0]  ==  0,  {C1}] // Flatten

C1 = C1 /. solv1

Clear[h1];  
    h1[{x_, y_}] := - x + 3 y  +  -9;   h1[r]

d1 = C1/vecLen[n]

N[d1]

d = -6;  d2 = d + d1

N[d2]

n1 = en * d2 // Simplify

N[n1]

r2[t_] :=  n1  +  a t;  r2[t]

solvX = Solve[r2[t][[1]] == 0, {t}] // Flatten

r2[t] /. solvX  // Simplify

N[%]

solvY = Solve[r2[t][[2]] == 0, {t}] // Flatten

r2[t] /. solvY  // Simplify

N[%]

Ueb/Ex 2      W

OP1 =  {-5, 6}

r02 = OP1 - r0

h2[{x_, y_}] := - x + 3 y  +  C2 ;   solv2 = 
      Solve[h2[r02]  ==  0,  {C2}] // Flatten

C2 = C2 /. solv2

d2 = C2/vecLen[n]

N[%]

Ueb/Ex 3      W

Remove["Global`@*"]

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

r = {x, y, z};
r0 = {2, 4, 5};
a = {-4, 3, 0};
b = {0, -2, 5};
 Φ[λ_, μ_] := 
      r0 + λ a + μ b;   Φ[λ, μ]

solv3 = 
      Solve[Φ[λ, μ] == r, {λ, μ, 
            x}] // Flatten

coord = (x1 - x  /. solv3) //  Expand

n = {1, 4/3, 8/15};  C1 = -10;  n

en =  n/vecLen[n]

N[%]

d = C1/vecLen[n]

N[%]

r = {x, y, z}

Solve[{x0, 0, 0} . n + C1 == 0, {x0}] // Flatten

Solve[{0, y0, 0} . n + C1 == 0, {y0}] // Flatten

N[%]

Solve[{0, 0, z0} . n + C1 == 0, {z0}] // Flatten

N[%]

vol =  10*15/2 *75/4 /6

N[%]