Lösungen

1

A A|    (A^(-1)) (A^(-1))  M X A A  = (A^(-1))A (M^(-1)) A (A^(-1)) A + E     |(A^(-1))

A A  (A^(-1)) (A^(-1))  M X A A (A^(-1))  =A A  E (M^(-1)) E A (A^(-1))  +A A  E  (A^(-1))

(M^(-1)) |    M X A  =A A (M^(-1)) +A      |(A^(-1))

(M^(-1))M  X  =(M^(-1)) A A (M^(-1))  (A^(-1)) +(M^(-1)) E A (A^(-1))

X  =(M^(-1)) A A (M^(-1))  (A^(-1))  +(M^(-1))     oder

X  =(M^(-1))(A^2) (M^(-1)) (A^(-1)) +(M^(-1))= (M^(-1))((A^2) (M^(-1)) (A^(-1)) +E)

2

rem; ove;   Remove["Global`*"]

OA1 = {-1, 1, 1};   A1B1 = {2, 1, 1};   A1C1 = {4, 
        5, -2};
OA2 = {1, -1, 2};   A2B2 = {1, 2, -1};   A2C2 = {-3, 
        2, -4};
OQ1 = {10, -10, 12};   OQ2 = {-5, -6, 
        8};
v1[λ_, μ_] := 
        OA1 + λ  A1B1 + μ A1C1;
v2[ν_, σ_] := 
        OA2 +  ν  A2B2 + σ  A2C2;

a

VTetraeder = Det[ {OA2 + A2B2 - OA1, A1C1, OQ2 - OA1}]/6

N[%]

b

distQ2Φ2 = 
      Det[{ OQ2 - OA2,  A2B2,  A2C2}]/Norm[Cross[A2B2,  A2C2]]

N[%]

c

solv01 = 
      Solve[v1[λ, μ] == 
            v2[ν, σ], {μ, ν, σ}] // 
        Flatten

s1[λ_] := v1[λ, μ] /. solv01;  
    s1[λ] // Simplify

s2[λ_] := v2[ν, σ] /. solv01;  
    s2[λ] // Simplify

solv02 = 
      Solve[s1[λ][[1]] == 0, {λ}] // 
        Flatten

OD1 = s1[λ] /. solv02

N[%]

solv03 = 
      Solve[s1[λ][[2]] == 0, {λ}] // 
        Flatten

OD2 = s1[λ] /. solv03

N[%]

solv04 = 
      Solve[s1[λ][[3]] == 0, {λ}] // 
        Flatten

OD3 = s1[λ] /. solv04

N[%]

d

gQ1Q2[t_] := OQ1 +  t (OQ2 - OQ1);  gQ1Q2[t]

solv05 = 
      Solve[v1[λ, μ] == 
            v2[ν, σ], {μ, ν, σ}] // 
        Flatten

solv05 = 
      Solve[v1[λ, μ] == gQ1Q2[t], {λ, μ, t}] // 
        Flatten

N[%]

OQ3 = gQ1Q2[t] /. solv05

N[%]

solv06 = 
      Solve[v2[ν, σ] == gQ1Q2[t], {t, ν, σ}] // 
        Flatten

N[%]

OQ4 = gQ1Q2[t] /. solv06

N[%]

Norm[OQ4 - OQ3]

N[%]

e

winkel = ArcCos[OQ1 . OQ2/(Norm[OQ1] Norm[OQ2])]

N[%]

%/Degree

3

Remove["Global`*"]

M = {{2, -2, 3}, {-3, 1, 3}, {-2, -1, 
            2}};
M // MatrixForm

a

Det[M]

Det[M + M]

2^3 Det[M]

b

Det[M] ist nicht 0, also existiert die Inverse

c

invM = Inverse[M]

invM // MatrixForm

invM // N // MatrixForm

d

transpInvM = Inverse[Transpose[M]]

Transpose[Inverse[M]]

transpInvM // MatrixForm

transpInvM  //  N  // MatrixForm

e

{invM // MatrixForm,  transpInvM // MatrixForm}

Transpose[invM] ==  transpInvM

f

Det[invM . invM . invM . invM . invM . invM . invM]

N[%]

Det[invM . invM . invM . invM . invM . invM . invM] == 
      Det[invM]^7

g

OP0 = {2, 5, 8};   OP1 = M . OP0

OP2 = invM . OP0

N[%]

h

M2 = M . M;  M2 // MatrixForm

M2 . OP2 == OP1

M2 = M.M

4

Remove["Global`*"]

a

ga1 = (3  x + 3  y + 4  z == 
            2);
ga2 = (x + 2  y - z == 1);
ga3 = (4  x + 2  y + 10  z == 
            2);
Solve[{ga1, ga2, ga3}, {x, y, z}] // Flatten

Solve[{ga1, ga2, ga3}, {x, y}] // Flatten

N[%] // ExpandAll

Solve[{ga1, ga2, ga3}, {y, z}] // Flatten

N[%] // ExpandAll

Solve[{ga1, ga2, ga3}, {x, z}] // Flatten

N[%] // ExpandAll

b

gb1 = ga1;
gb2 = ga2;
gb3 = (4  x + 2  y + 10  z == 
            5);
Solve[{gb1, gb2, gb3}, {x, y, z}] // Flatten

5

Remove["Global`*"]

OO = {0, 0, 0};
OA1 = OO;   OA2 = {4, -2, -1};   OA4 = {1, 5, 2}; 
    OB1 = {2, 1, 10};  OC1 = OB1;

a

OA3 = OA2 + (OA4 - OA1)

OB2 = OB1 + (OA2 - OA1)

OB3 = OB2 + (OA3 - OA2)

OB4 = OB1 + (OA4 - OA1)

OC2 = OA2 + 0.7 (OB2 - OA2)

OC4 = OA4 + 0.8 (OB4 - OA4)

OC3 = OC2 + (OC4 - OC1)

b

vSpat = Det[{OA2 - OA1, OA3 - OA1, OB1 - OA1}]

c

vKaese1 = Abs[Det[{OB1 - OB4, OB3 - OB4, OC4 - OB4}]]/2

vKaese2 = Abs[Det[{OB1 - OB2, OC2 - OB2, OC3 - OC2}]]/2

vAbgeschnitten = vKaese1 + vKaese2

d

vAbgescProzent = vAbgeschnitten/vSpat *100

6

Remove["Global`*"]

OA[1] = {-1, 0, 1};   OA[2] = {4, -2, -1};   OA[
        3] = {1, 5, 2}; OA[4] = {2, 1, 10};  
    M = {{2, -2, 3}, {-3, 1, 3}, {-2, -1, 2}};   M //  
      MatrixForm

a

vSpat = Det[{OA[2] - OA[1], OA[3] - OA[1], OA[4] - OA[3]}]

grundFlaeche = Norm[Cross[OA[2] - OA[1], OA[4] - OA[3]]]

abstand = vSpat/grundFlaeche

N[%]

hVecSenkr = 
      Cross[OA[2] - OA[1], OA[4] - OA[3]]/grundFlaeche abstand

g1[t_] := OA[1] + t (OA[2] - OA[1]);
g2[s_] := OA[3] + s (OA[4] - OA[3]);
solv6 = 
      Solve[g1[t] + c hVecSenkr  == g2[s], {t, s, c}] // 
        Flatten

OP1 = g1[t] /. solv6

N[%]

OP2 = g2[s] /. solv6

N[%]

(g1[t] + c hVecSenkr  == g2[s]) /. solv6

b

vTetraeder = 
      Det[{OA[2] - OA[1], OA[3] - OA[1], OA[4] - OA[1]}]/6

c

B[k_] := M . OA[k];
ta = Table[B[k], {k, 1, 4}]

ta // Transpose // MatrixForm

Det[{B[2] - B[1], B[3] - B[1], B[4] - B[1]}]

Det[{B[2] - B[1], B[3] - B[1], B[4] - B[1]}]/6