Lösungen

Remove["Global`*"]

Problem 1

vecLen[r_] := Sqrt[r . r]

r01 = {2, 1, 3};  r02 = {-3, -1, 1};  a = {1, -2, 1}; 
    b = {-2, 1, 4};  r = {x, y, z};

rΦ[{λ_, μ_}] :=  
      r01  +  1/
            2 (r02 - r01)  + λ a  +  μ b

n = Cross[a, b]

Φ[{x_, y_, z_}] :=  
      n . {x, y, z}/vecLen[n] +  C1;  Φ[r]

en = n/vecLen[n]

N[%]

solv1 = 
      Solve[Φ[r01  +  1/2 (r02 - r01)] == 0, {C1}] // 
        Flatten

Φ[r] /. solv1

N[%] // Expand

Problem 2

a = {5, 2, 1};  b = {1, 4, 8};  c = {-2, y, 4}

V  =  Det[{a, b, c}]

Solve[V == 100, {y}] // Flatten

N[%]

Problem 3

a = {2, 1, 3}; b = {-3, -1, 1}; c = {5, 2, 1};

d1 = Cross[Cross[a, b], c]

d2 = Cross[a, Cross[b, c]]

Cross[d1, d2]

Problem 4

d0 = 
      Det[{{4, -2, 3}, {8, -2, 
            4}, {9, α, β}}]

Solve[{
4  x - 2  y + 3  z - 5 == 
            0, 
8  x - 2  y + 4  z - 6 == 
            0, 
9  x + α y + β z + 
              4 == 0}, {x, y, z}]  // Flatten

Repetitionsaufgaben

Remove["Global`*"]

Problem 5

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

% // ExpandAll

% // N

Problem 6

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}]}]];

Problem 7

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

Problem 8

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

Problem 9

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[%]

Arbeiten mit Matlab oder Octave sowie CAS

Lösung Aufgabe 1 mit MatLab

>> a1=[0 0 0 -1 1 0 0 0 0 (50+1)]                                                                                                                        
a1 =

         0         0         0        -1         1         0         0         0         0        51

>> a2=[0 -1 1 -1 0 0 0 0 0 20]
a2 =

         0        -1         1        -1         0         0         0         0         0        20

>> a3=[-1 1 0 0 0 0 0 0 0 (50-10)]
a3 =

        -1         1         0         0         0         0         0         0         0        40

>> a4=[0 0 0 0 1 1 0 0 0  50+56]
a4 =

        0        0        0        0        1        1        0        0        0      106

>> a5=[0 0 1 0 0 1 -1 -1 0 0] 
a5 =

         0         0         1         0         0         1        -1        -1         0         0

>> a6=[-1 0 0 0 0 0 1 0 -1 100]
a6 =

        -1         0         0         0         0         0         1         0        -1       100

>> a7=[0 0 0 0 0 0 0 1 1 10+5]
a7 =

        0        0        0        0        0        0        0        1        1       15

>> m=[a1;a2;a3;a4;a5;a6;a7]
m =

         0         0         0        -1         1         0         0         0         0        51
         0        -1         1        -1         0         0         0         0         0        20
        -1         1         0         0         0         0         0         0         0        40
         0         0         0         0         1         1         0         0         0       106
         0         0         1         0         0         1        -1        -1         0         0
        -1         0         0         0         0         0         1         0        -1       100
         0         0         0         0         0         0         0         1         1        15

>> rref(m)
ans =

         1         0         0         0         0         0        -1         0         1      -100
         0         1         0         0         0         0        -1         0         1       -60
         0         0         1         0         0         1        -1         0         1        15
         0         0         0         1         0         1        -0        -0        -0        55
         0         0         0         0         1         1         0         0         0       106
         0         0         0         0         0         0         0         1         1        15
         0         0         0         0         0         0         0         0         0         0
         
Transponieren um besser  lesen zu können:

>> rref(m)'
ans =

         1         0         0         0         0         0         0
         0         1         0         0         0         0         0
         0         0         1         0         0         0         0
         0         0         0         1         0         0         0
         0         0         0         0         1         0         0
         0         0         1         1         1         0         0
        -1        -1        -1        -0         0         0         0
         0         0         0        -0         0         1         0
         1         1         1        -0         0         1         0
      -100       -60        15        55       106        15         0

Mit einem andern Programm

a1={0, 0, 0, -1, 1, 0, 0, 0, 0};
a2={0, -1, 1, -1, 0, 0, 0, 0, 0};
a3={-1, 1, 0, 0, 0, 0, 0, 0, 0};
a4={0, 0, 0, 0, 1, 1, 0, 0, 0};
a5={0, 0, 1, 0, 0, 1, -1, -1, 0};
a6={-1, 0, 0, 0, 0, 0, 1, 0, -1};
a7={0, 0, 0, 0, 0, 0, 0, 1, 1};
m={a1,a2,a3,a4,a5,a6,a7};
u=RowReduce[m];
Print[u//MatrixForm];
b={51,20,40,106,0,100,15};
Print[RowReduce[Append[Transpose[m],b]]//MatrixForm];
x={x1,x2,x3,x4,x5,x6,x7,x8,x9};
Solve[m.x==b,x] //Flatten

x6, x7, x9 sind hier Parameter!

Prepend[Transpose[m],b];

Nochmals

Solve[m.x==b,{x1,x2,x3,x4,x5,x8}] //Flatten

x6, x7, x9 sind hier Parameter!