Lösungen

1

Remove["Global`*"]

M . (Id - X) . Inverse[M] + M - A . M == 
      A . Transpose[M] - 3 M

Id - M . X . Inverse[M] + M - A . M == 
      A . Transpose[M] - 3 M

Id - A . Transpose[M] + 3 M - A . M  +  M == 
      M . X . Inverse[M]

X == Inverse[M] . (Id - A . Transpose[M] + 4 M - A . M) . 
        M

X == Id - Inverse[M] . A . Transpose[M] . M + 4 M - 
        Inverse[M] . A . M . M

2

Remove["Global`*"]

r0 = {1, 2, -1};
a = {2, 1, 1};
q = {3, 10, 14};
rq = q - r0

gq = Norm[Cross[a, rq]]/Norm[a]

N[%]

3

Remove["Global`*"]

r0 = {1, 2, -1};
a = {2, 1, 1};
b = {1, -1, -2};
q = {3, 10, 14};
rq = q - r0

a

gq = -Det[{a, b, rq}]/Norm[Cross[a, b]]

N[%]

b

v[λ_, μ_] := r0 + λ a + μ b;  
    v[λ, μ ]

w[t_] := q + t Cross[a, b];  w[t]

solv1 = 
      Flatten[Solve[ 
          v[λ, μ ] == w[t], {λ, μ , t}]]

t0 = t /. solv1

w[t0]

N[%]

4

Remove["Global`*"]

p1 = {1, 1, 0};  p2 = {-1, 2, 2}; p3 = {-3, -2, 3}; 
    p4 = {1, 1, 4};
m = {v1 = p2 - p1, v2 = p3 - p2, v3 = p4 - p3}

m + 1/2 m

s = Sum[1/2^k, {k, 0, 99}]

s // N

s m

s m // N

TotalPunkte = (3*100 + 1) Punkte

p301 = p1 + Apply[Plus,  s m]  // N

5

Remove["Global`*"]

Solve[{x + y + z == 1, x - y + z == 1, 
        x - y - z == 0}, {x, y, z}]

6

Remove["Global`*"]

A = {{1, 2, 3}, {2, 3, 1}, {0, 1, 1}};  A // MatrixForm

B = {{1, 0, 3}, {2, 3, 1}, {0, 1, 1}}; A//MatrixForm

A . B // MatrixForm

Inverse[A] // MatrixForm

Inverse[A] // N // MatrixForm

Inverse[B] // MatrixForm

Inverse[B] // N // MatrixForm

Transpose[Inverse[A . B]] // MatrixForm

N[%] // MatrixForm

Transpose[Inverse[B] . Inverse[A]] == 
      Transpose[Inverse[A . B]]

7

M = {{1, -1}, {2, 1}}; p0 =  {4, 7};  M // MatrixForm

p1 = M . p0

α = 32 Degree;  
    d32 = {{Cos[α], -Sin[α]}, {Sin[α], 
          Cos[α]}};  d32 // N // MatrixForm

p2 = d32 . p1

p2 // N

p3 = M . p2

p3 // N

8

Remove["Global`*"]

p1 = {1, 1, 0};  p2 = {1, 0, 2}; p3 = {0, 2, 3};

S = 1/3 (p1 + p2 + p3)

N[%]

Vp1p2p3 = 1/2 Norm[Cross[p2 - p1, p3 - p1]]

N[%]

p[t_] := t S;

norm[v_] := Sqrt[v . v]

Vof[t_] := 
      1/2 (norm[Cross[p2 - p1, p[t] - p1]] + 
            norm[Cross[p3 - p2, p[t] - p2]] + 
            norm[Cross[p1 - p3, p[t] - p3]]);   Vof[t] // 
      ExpandAll

solv = Solve[2  Vp1p2p3 == Vof[t], {t}] // N // 
        Flatten

Plot[Vof[t], {t, -1, 3}]

p[t] /. solv[[2]]

9

Remove["Global`*"]

r1 = {1, 2, -1};   r2 = {-1, 1, 3};   a = {2, 1, 
        1};   b = {3, -1, 2};

v1[t_] := r1 + t a;  v2[r_] := r2 + r b;

a

V = Det[{a, b, r1 - r2}]

Determinante nicht 0, nicht windschief

b  (Sprachproblem: was ist der negative oder der positive Fall?)

d = V/Norm[Cross[a, b]]

N[%]

10

Remove["Global`*"]

a, b

r = 2;  k[x_, y_, z_] := x^2 + y^2 + z^2 - r^2

ϕ[x_, y_, z_, a_] := x + y + z - a

ϕ[{x_, y_, z_}, a_] := ϕ[x, y, z, a]

k[x, x, x] == 0

solv = Solve[  k[x, x, x] == 0, {x}] // Flatten

TPunkt = {(2/(Sqrt[3])), (2/(Sqrt[3])), (2/(Sqrt[3]))}

N[%]

ϕ[TPunkt, a] == 0

Solve[ϕ[TPunkt, a] == 0, {a}]

c

e1 = {1, 0, 0};  e2 = {0, 1, 0};  e3 = {0, 0, 1}; 
    T =  TPunkt;

α = ArcCos[T . e1/Norm[T] / Norm[e1]]

β = ArcCos[T . e2/Norm[T] / Norm[e2]]

γ = ArcCos[T . e3/Norm[T] / Norm[e3]]

N[%]

% /Degree