Lösungen

1

F normiert

F = {2, 5, 6};  30 F/Norm[F]

% // N

solv = 
      Solve[30 F/
                Norm[F] ==  λ {1, 1, 0} +  μ {0, 1, 
                  1} +  ν {1, 1, 1} , {λ, μ, ν}]  // 
        Flatten

% // N

Die Streckungen

Die drei Lösungsvektoren

{λ {1, 1, 0},  μ {0, 1, 1},  ν {1, 1, 1}}  /. 
        solv   // MatrixForm

% // N   // MatrixForm

2

Normalenvektoren

n1 = {2, 3, -1};  n2 = Cross[{1, 1, 1}, {0, 2, 3}]

Winkel zwischen den Normalenvektoren

ArcCos[n1 . n2/(Norm[n1] Norm[n2])]

%//N

In Grad

%/Degree

180 - %

3

Spiegelungsgerade:

v[λ_] := {4, 4, 6} + λ {2, 3, -1}

x[λ_] := v[λ][[1]];
y[λ_] := v[λ][[2]];
z[λ_] := v[λ][[3]];

{x[λ], y[λ], z[λ]}

Schnittpunkt:

solv = 
      Solve[2  x[λ] + 3  y[λ] - z[λ] == 
            2, {λ}] // Flatten

sVec = v[λ] /. solv

N[%]

P'

OP = {4, 4, 6};

OP' = OP + 2 (sVec - OP)

N[%]

Abstand von der Geraden:

d = 
      Norm[Cross[(OP' - {0, 1, 1}), {1, 1, 1}]]/
        Norm[{1, 1, 1}]

N[%]

4

Kreis

k[x_, y_, 
        z_] := ((x - 2)^2 + (y - 1)^2 + (z - 3)^2 == 
          25)

k[x, y, 0]

k[2, y, 0]

Solve[k[2, y, 0], {y}] // Flatten

Mittelpunkt

mM = {2, 1, 3}

x Mittelpunkt stimmt mit x = 2 auf dem Kreis überein. Daher gehen die beiden möglichen Gerade durch {2, -3} oder {2, 5} und sind parallel zur x-Achse. Die y-Koordinaten sind -3 und 5.

5

Plot

v[u_] := {u, u, -u^2 + 9};

ParametricPlot3D[v[u], {u, -3, 3}];

Tangentenvektor

D[v[u], {u}]

v[u_] = {u, u, -u^2 + 9}; D[v[u], {u}]

ParametricPlot3D[
        Evaluate[D[v[u], {u}]], {u, -3, 3}];

Länge

Länge  =  
      Integrate[Evaluate[Norm[D[v[u], {u}]]], {u, -3, 3}]

N[%]

Begleitendes Dreibein

a[u_] := D[v[u], {u}]/Norm[Evaluate[D[v[u], {u}]]];  a[u]

b = {1, -1, 0}/Norm[{1, -1, 0}]

c[u_] :=  Cross[a[u], b];  c[u]

Schlauch

schlauch[u_, t_] :=  
        v[u] + Cos[t] b + Sin[t] c[u]; 
schlauch[u, 1] /. u -> 1

schlauch[u, t]

schlauch[u_, 
        t_] :=  {u + 
            Cos[t]/(Sqrt[2]) - (((Sqrt[2]) u Sin[t])/(Sqrt[2 + 4 Abs[u]^2])), 
          u - Cos[t]/(Sqrt[2]) - (((Sqrt[2]) u Sin[t])/(Sqrt[2 + 4 Abs[u]^2])), 
          9 - (u^2) - (((Sqrt[2]) Sin[t])/(Sqrt[2 + 4 Abs[u]^2]))} /. 
        Abs[u]^2 -> u^2

schlauch[u, t]

ParametricPlot3D[
        schlauch[u, t], {u, -3, 3}, {t, 0, 2 Pi}];

Fläche

Sqrt[
      Cross[Evaluate[D[schlauch[u, t], {u}]],       Evaluate[
            D[schlauch[u, t], {t}]]] . 
        Cross[Evaluate[D[schlauch[u, t], {u}]],       Evaluate[
            D[schlauch[u, t], {t}]]]]

√(((((((Sqrt[2]) Cos[t])/(Sqrt[2 + 4 (u^2)])) + ((2 (Sqrt[2]) (u^2) Cos[t])/(Sqrt[2 + 4 (u^2)])) + (Sqrt[2]) u Sin[t] - ((2 Cos[t] Sin[t])/(2 + 4 (u^2))) - ((4 u Sin[t]^2)/(((2 + 4 (u^2)))^(3/2)))))^2) + (((-(((Sqrt[2]) Cos[t])/(Sqrt[2 + 4 (u^2)])) - ((2 (Sqrt[2]) (u^2) Cos[t])/(Sqrt[2 + 4 (u^2)])) + (Sqrt[2]) u Sin[t] + ((2 Cos[t] Sin[t])/(2 + 4 (u^2))) - ((4 u Sin[t]^2)/(((2 + 4 (u^2)))^(3/2)))))^2) + ((((Sqrt[2]) Sin[t] + ((8 (u^2) Sin[t]^2)/(((2 + 4 (u^2)))^(3/2))) - ((2 Sin[t]^2)/(Sqrt[2 + 4 (u^2)]))))^2)) /. {t -> h2, u -> h1}

NIntegrate[√(((((((Sqrt[2]) Cos[h2])/(Sqrt[2 + 4 (h1^2)])) + ((2 (Sqrt[2]) (h1^2) Cos[h2])/(Sqrt[2 + 4 (h1^2)])) + (Sqrt[2]) h1 Sin[h2] - ((2 Cos[h2] Sin[h2])/(2 + 4 (h1^2))) - ((4 h1 Sin[h2]^2)/(((2 + 4 (h1^2)))^(3/2)))))^2) + (((-(((Sqrt[2]) Cos[h2])/(Sqrt[2 + 4 (h1^2)])) - ((2 (Sqrt[2]) (h1^2) Cos[h2])/(Sqrt[2 + 4 (h1^2)])) + (Sqrt[2]) h1 Sin[h2] + ((2 Cos[h2] Sin[h2])/(2 + 4 (h1^2))) - ((4 h1 Sin[h2]^2)/(((2 + 4 (h1^2)))^(3/2)))))^2) + ((((Sqrt[2]) Sin[h2] + ((8 (h1^2) Sin[h2]^2)/(((2 + 4 (h1^2)))^(3/2))) - ((2 Sin[h2]^2)/(Sqrt[2 + 4 (h1^2)]))))^2)), {h1, -3, 3}, {h2, 0, 2 Pi}]