Lösungen/ Solutions
Vorversion /Version préliminaire

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

Aufgabe /Problème 1

OP1 = {0, 0, 0}; OP2 = {5, 5, 5}; OP3 = {5, 7, 1}; 
    OP4 = {-3, -4, -6};

Det[{OP3 - OP4, OP2 - OP1, OP1 - OP4}]

vecLen[Cross[OP3 - OP4, OP2 - OP1]]

N[%]

d = 
      Det[{OP3 - OP4, OP2 - OP1, OP1 - OP4}]/
          vecLen[Cross[OP3 - OP4, OP2 - OP1]]   //  Abs

N[%]

r = d/2

N[%]

Aufgabe /Problème 2

Remove["Global`*"]; vecLen[a_] := Sqrt[a . a]

OP1 = {0, 0, 0}; OP2 = {1, 1, 1}; OP3 = {4, 6, 3}; 
    OP4[z_] := {2, 3, z};

Det[{OP2 - OP1, OP3 - OP1, OP4[8] - OP1}]/6

N[%]

Solve[
      Det[{OP2 - OP1, OP3 - OP1, OP4[z] - OP1}]/6 == 
        20, {z}]

N[%]

Aufgabe 3 /Problème 3

g1[x_, y_, z_] := x + y + z == 1;
g2[x_, y_, z_, α_] := 
        x + 2  y + 3  z == α;
g3[x_, y_, z_, β_] := 
        x + 4  y + β z == 0;

Solve[{g1[x, y, z], g2[x, y, z, α], g3[x, y, z, β]}, {x, 
        y, z}]

d0 = Det[{{1, 1, 1}, {1, 2, 4}, {1, 3, β}}]

d1 = Det[{{1, α, 0}, {1, 2, 4}, {1, 3, β}}]

d2 = Det[{{1, 1, 1}, {1, α, 0}, {1, 3, β}}]

d3 = Det[{{1, 1, 1}, {1, 2, 4}, {1, α, 0}}]

{d1/d0, d2/d0, d3/d0}

{d1/d0, d2/d0, d3/d0} /. α -> 0

===>     β=7

{d1/d0, d2/d0, d3/d0} /. β -> 9

Solve[d1/d0 == 0, {α}] /.  β -> 9

N[%]

Aufgabe 4 /Problème 4

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

A = vecLen[Cross[{5, 0, 1}, {1, 5, 4}]]

N[%]

n1 = Cross[{5, 0, 1}, {1, 5, 4}]

n2 = Cross[{2, 1, -8}, {1, 5, 4}]

ϕ = ArcCos[n1 . n2/vecLen[n1]/vecLen[n2]]

N[%]

%  /Degree