Lösungen

1.

Remove["Global`*"]

a

Solve[{x + 2  y + 3  z == 4,  
        x + 4  y + 9  z == 
          16, λ x + 6  y + 12  z == μ}, {x, y, 
        z}]

Solve[{x + 2  y + 3  z == 4,  
        x + 4  y + 9  z == 16, λ x + 6  y + 12  z == 
          20}, {x, y, z}]

Fall unmöglich.

b

Solve[{x + 2  y + 3  z == 4,  
        x + 4  y + 9  z == 16, λ x + 6  y + 12  z == 
          1}, {x, y, z}]

Solve[{x + 2  y + 3  z == 4,  x + 4  y + 9  z == 16, 
        2 x + 6  y + 12  z == 1}, {x, y, z}]

λ = 2

c

Solve[{1 + 2*1 + 3  z == 4,  
        1 + 4*1 + 9  z == 16, λ *1 + 6*1 + 12  z == 
          1}, {z, λ, μ}]

2.

Remove["Global`*"]

OA  =  {0, 1, 1};  OB  =  {1, 1, 2};  
    OC[λ_] := λ {1, -1, 0};

Vabs = 10;  Det[{OA, OB, OC[λ]}]/6

a

Fall unmöglich.

b

OM  =  (OA + OB + OC[λ])/3

OA + 1/2 (OB - OA) + 
        1/3 (OC[λ] - (OA + 1/2 (OB - OA))) // 
      Simplify

OP  =  3 OM

c

Det[{OA, OB, OC[λ]}]/6

Det[{(OP - OA), (OP - OB), (OP - OC[λ])}]/
      6

Det[{(OP - OA), (OB), (OC[λ])}]/
      6

Det[{(OA), (OP - OB), (OC[λ])}]/
      6

Det[{(OA), (OP - OB), (OC[λ])}]/
      6

Det[{(OA), (OP - OB), (OP - OC[λ])}]/6

Det[{(OP - OA), (OB), (OP - OC[λ])}]/6

Det[{(OP - OA), (OP - OB), (OC[λ])}]/6

Inhalte immer 0

3.

Remove["Global`*"]

a

{x - 2, y - (-1)} . {x - 10, y - 6} == 0

Sqrt[{x - 2, y - (-1)} . {x - 2, y - (-1)}] == 
      3

Solve[{{x - 2, y - (-1)} . {x - 10, y - 6} == 0, 
        Norm[{x - 2, y - (-1)}] == 3}, {x, y}]

N[%]

Remove["Global`*"]

a

(x - 2)^2 + (y - (-1))^2 == 3^2

OP = {10, 6}

(y - 6)/(x - 10) == k

Solve[{(x - 2)^2 + (y - (-1))^2 == 
          3^2, (y - 6)/(x - 10) == k}, {x, y}]

solv = 
      Solve[(Sqrt[-40 + 112 k - 55 (k^2)]) == 0, {k}] // 
        Flatten

f1[x_] := k x - k 10  +  6 /. solv[[1]]

f1[x]

f2[x_] := k x - k 10  +  6 /. solv[[2]]

f2[x]

solv1 = 
      Solve[(x - 2)^2 + (f1[x] - (-1))^2 == 3^2, {x}] // 
        Flatten

% // N

solv2 = 
      Solve[(x - 2)^2 + (f2[x] - (-1))^2 == 3^2, {x}] // 
        Flatten

% // N

OT1 = {x, f1[x]} /. solv1[[1]]

% // N

OT2 = {x, f2[x]} /. solv2[[1]]

% // N

b

Inh  =  Det[{OT2 - OP, OT1 - OP}]/2

% // N

c

Remove["Global`*"]

OM = {2, -1};

OP = {10, 6}

d[λ_] := OM +  λ (OP - OM)/Norm[ (OP - OM)]; 
    d[λ]

d[3]

% // N

4.

Remove["Global`*"]

a

g1[λ_] := {0, 1, 
            1} + λ {4, 3, 2};
g2[μ_] := {1, 3, 
            2} + μ {0, -1, 1};
OP = {10, 6, 1};

h1 = Norm[Cross[OP - {0, 1, 1}, {4, 3, 2}]]/Norm[{4, 3, 2}]

% // N

h1 = 
      Norm[Cross[OP - {1, 3, 2}, {0, -1, 1}]]/
        Norm[{0, -1, 1}]

% // N

g1  liegt näher

b

dist = 
      Norm[Det[{{1, 3, 2} - {0, 1, 1}, {4, 3, 2}, {0, -1, 1}}]]/
        Norm[Cross[{4, 3, 2}, {0, -1, 1}]]

% // N

c

r = dist/2

% // N