Lösungen

1

a

ans =
  -1.41421
    1.00000

b

ans = 0.00019000

c

Inf

d

ans = 0   -3       2

e

ans =

  1.0e+01  *

   3.40000
   2.90000
   6.00000
  -4.76000
   8.00000

f

ans = 8.9000

2

Remove["Global`*"]

a

u = (3  a x  +  2 y  +  3 z == 
            1);
v = (2 x  + 2 a y  +  4 z == 
            1);
w = (3 x  +  4 y  + 1 a z == 
            1);
Solve[{u, v, w}, {x, y, z}]

Solve[24 - 35 a + 3 (a^3) == 0, {a}] // N

b

Solve[{u, v, w}, {x, y, z}] /. a -> 1

c

v1 = {a 3, 2, 3}; v2 = {2, a 2, 4}; 
    v3 = {3, 4, a 1};
v1 - v2 + 3  v3

Det[{v1, v2, v3}] /. a -> 3

Keine Dim. da keine Lösung

Solve[{u, v, w}, {x, y, z}] /. a -> 3

d

Solve[{u, v}, {x, y, z}] /. a -> 2

g[t_] := {(1/10) - (t/5), (1/5) - ((9 t)/10), t};  
    P0 = {1, 1, 1};
v = g[1] - g[0];  u  =  P0 - g[0];
d = Norm[Cross[v, u]]/Norm[v]

N[%]

3

a

v1 = {3, 2, 3}; v2 = {2, 2, 4};  v3 = {3, 4, 1};  
    w = {10, 12, 2};
Det[{v1, v2, v3}]

b

Solve[
      w == λ1 v1 + λ2 v2 + λ3 v3, {λ1, λ2, λ3}]

4

Remove["Global`*"]

a

a = {10, 10, 10};  e1 = {1, 0, 0};  e2 = {0, 1, 0}; 
    e3 = {0, 0, 1};
LaengeHalbdiagonale = Norm[a]

N[%]

Winkel Diagonale-Achse

WinkelDiagonaleAchse = 
      ArcCos[a . e1/(Norm[a]  Norm[e1])]

N[%]

α = N[%]/Degree

Test mit Richtungscosinusen o.k.:

solv = Solve[3 Cos[x]^2 == 1, {x}] // Flatten

solv4 = x /. solv[[4]]

WinkelDiagonaleAchse == solv4

Skizze Würfel:

P1 = {1, 1, 1};  P2 = {-1, 1, 1};  P3 = {-1, -1, 1};  
    P4 = {1, -1, 1}; 
P5 = {1, 1, -1};  P6 = {-1, 1, -1};  
    P7 = {-1, -1, -1};  
    P8 = {1, -1, -1}; 
x1 = {1, 0, 0}; x2 = {-1, 0, 0}; y1 = {0, 1, 0}; 
    y2 = {0, -1, 0}; z1 = {0, 0, 1}; 
    z2 = {0, 0, -1};
linX = Line[{x1, x2}];
linY = Line[{y1, y2}];
linZ = Line[{z1, z2}];
linO1 = 
        Line[{{0, 0, 0}, {-0.75, 1, 1}}];
lin02 = 
        Line[{{0, 0, 0}, {1, 1, -0.75}}];
lin03 = 
        Line[{{0, 0, 0}, {1, -0.75, 1}}];
lin04 = Line[{{0, 0, 0}, P1}];
linP = {PointSize[0.03], Point[P1], Point[{-0.75, 1, 1}], 
          Point[{1, 1, -0.75}], Point[{1, -0.75, 1}], 
          Point[{0, 0, 0}], 
          Line[{P1, P2, P3, P4, P1, P5, P6, P2, P6, P7, P3, P7, P8, P4, P8, 
              P5}], linX,  linY,  linZ, linO1, lin02, lin03, 
          lin04};
Show [
        Graphics3D[
          linP ], 
 (*ViewPoint -> {-1.424,  4.258,  
              2.660}*) 
ViewPoint -> {0.614,  -1.560,  2.660}, Boxed -> False];

Winkel zwischen zwei Diagonalen:

WinkelDiagonaleDiagonale = 
      ArcCos[P1 . P2/(Norm[P1] Norm[P2])]

N[%]

β = N[%]/Degree

Winkel Kanten-Diagonalen:

ArcCos[P1 . (P1 - P2)/(Norm[P1] Norm[P1 - P2])]

N[%]

δ = N[%]/Degree

Nochmals Winkel Kanten-Diagonalen:

γ = (180 - β)/2

WinkelKanteDiagonale = 
      Pi - ArcCos[
          P1 . (P2 - P1)/(Norm[P1] Norm[P2 - P1] )]

N[%]

δ = N[%]/Degree

b

Winkel Kante Achse

Diagonalwinkel mit Achse :   ArcCos[(1/(Sqrt[3]))],  
    Winkel Kanten - Diagonalen : π - 
          ArcCos[-(1/(Sqrt[3]))] oder ArcCos[(1/(Sqrt[3]))]

{π - ArcCos[-(1/(Sqrt[3]))], ArcCos[(1/(Sqrt[3]))]} // N

WinkelKanteAchse = 
      Pi - WinkelKanteDiagonale - WinkelDiagonaleAchse

N[%]

WinkelKanteAchse = 
      Pi - 2 WinkelDiagonaleAchse  // Simplify

N[%]

N[%]/Degree

(Sin[WinkelKanteAchse]) // N

LaengeKanteAchse = LaengeHalbdiagonale/ 
          Sin[WinkelKanteAchse]  * Sin[
          WinkelDiagonaleAchse]

N[%]

VolumenEckstueck1 = LaengeKanteAchse^3/6

N[%]

VKoerper = 2 VolumenEckstueck1

N[%]

Andere Variante

v1[t_] :=  a - t e1;  v2[t_] :=  a - t e2;  
    Solve[v1[t] . v2[t] == 0, {t}]

v1[t] . v2[t]

v1[15]

v2[15]

Norm[v1[15]]

{15  e1,  15 e2,  15 e3}

(15  e1 - a)  . (15 e2 - a)

(15  e1 - a)  . (15 e3 - a)

(15  e2 - a)  . (15 e3 - a)

VolumenEckstueck2 = 15^3/6

N[%]

VolumenKoerper = 2 VolumenEckstueck2

Kontrollen:

WinkelKanteAchse = ArcCos[v1[15] . e1/Norm[v1[15]]]

N[%]

N[%]/Degree

180 - %

Sin[% Degree]

Sin[%]

e[λ_] :=  λ e1;  
    e[λ] . (a - e[λ]) == 0

Solve[(10 - λ) λ == 
        0, {λ}]

c

kantenlaenge  =  2 LaengeHalbdiagonale/ Sqrt[3]

d

VolumenKoerper = 2 VolumenEckstueck2

5

Remove["Global`*"]

p1 = {4, 1};  p2 = {2, 3};  φ = 68.44 Degree;

a

m = {{Cos[φ], -Sin[φ]}, {Sin[φ], 
          Cos[φ]}};  m // MatrixForm

p3 = m . p2

b

Det[{p2 - p1, p2 + (p3 - p1)}]

6

Remove["Global`*"]

r1 = {1, 0, 1}; r2 = {1, 2, -1}; a1 = {4, -1, 1};  
    a2 = {-1, -1, 6};  p0 = {10, 1, -2};

a

Vol = Det[{a1, a2, r2 - r1}]

==> Schneiden sich nicht

Solve[a1 == α  a2, {α}]

==> windschief

b

d = Vol/(Norm[Cross[a1, a2]])

N[%]

c

d = Det[{a1, a2, p0 - r1}]/(Norm[Cross[a1, a2]])

N[%]

7

Remove["Global`*"]

p1 = {4, 1};  p2 = {2, 3}; p3 = {-6, -2};

a

k[x_, y_, r_] := ({x, y} - p1) . ({x, y} - p1) - 
          r^2;
k[vec_, r_] := (vec - p1) . (vec - p1) - r^2

k[p2, r]

Solve[k[p2, r] == 0, {r}]

N[%]

k[vec_] := k[vec, 2 (Sqrt[2])]

p4 = (p3 + p1)/2

r1 = Norm[p1 - p4]

N[%]

k1[vec_] := (vec - p4) . (vec - p4) - r1^2

{k[{x, y}] == 0, k1[{x, y}] == 0}

Solve[{k[{x, y}] == 0, k1[{x, y}] == 0}, {x, y}]

solv = N[%]

solv[[1]]

p5 = {x, y} /. solv[[1]];  T1 = p5

p6 = {x, y} /. solv[[2]];  T2 = p6

<< Graphics`ImplicitPlot`

pl = 
        ImplicitPlot[{k[{x, y}] == 0, 
            k1[{x, y}] == 0 }, {x, -7, 7}];

Prepend[Map[Point, {p1, p2, p3, p4, p5, p6}], 
      PointSize[0.04]]

lin = 
        Show[Graphics[Line[{p5, p3, p1, p5, p1, p6, p3}]]];

gra = 
        Show[Graphics[
            Prepend[Map[Point, {p1, p2, p3, p4, p5, p6}], PointSize[0.04]]], 
          pl, AspectRatio -> Automatic];

Show[gra,  lin];

b

len = Norm[p3 - p6]

Norm[p3 - p5]

c

len^2 == (s + 1)*s

solv = Solve[len^2 == (s + 1)*s, {s}]

s = s /. solv[[2]]

s*(s + 1) == len^2