Lösungen Teil 1

1

OA = {3, 0, 4};  OB = {1, 1, 1};  
    OC = {-7, 5, 11};
Solve[(OB - 
            OA) == λ (OC - OA), {λ}]

Die Punkte liegen icht auf einer Geraden.

2

a

a1 = {1, 3, 0};  a2 = {6, 2, 0};
r1 = {2, -3, 1};  
    r2 = {-1, -4, 1}; 
Solve[a1 == λ a2, {λ}]

Richtungsvektoren nicht parallel

solv = 
      Solve[r1 + λ a1 == r2 + μ a2, {λ, μ}] // 
        Flatten

Schnittpunkt = r1 + λ a1 /. solv[[1]]

b

a1 = {4, 3, 0};  a2 = {-8, -6, 0};
r1 = {0, 0, 0};  r2 = {10, 6, 3}; 
Solve[a1 == λ a2, {λ}]

Richtungsvektoren parallel

solv = 
      Solve[r1 + λ a1 == r2 + μ a2, {λ, μ}] // 
        Flatten

Kein Schnittpunkt

c

a1 = {0, 0, 1};  a2 = {2, 0, -1};
r1 = {3, 0, 5};  r2 = {1, -3, 6}; 
Solve[a1 == λ a2, {λ}]

Richtungsvektoren nicht parallel

solv = 
      Solve[r1 + λ a1 == r2 + μ a2, {λ, μ}] // 
        Flatten

Kein Schnittpunkt

d

a1 = {7, 2, -1};  
    a2 = {-14, -4, 2};
r1 = {4, -3, 2};  
    r2 = {-10, -7, 4}; 
Solve[a1 == λ a2, {λ}]

Richtungsvektoren parallel

solv = 
      Solve[r1 + λ a1 == r2 + μ a2, {λ, μ}] // 
        Flatten

3

xg[s_] := -1 + s  6;
yg[s_] := -4 + s  2;
zg[s_] :=    1 + s  0;
g[s_] := {xg[s], yg[s], zg[s]};    g[s]

xh[t_] :=   2 + t  1;
yh[t_] := -3 + t  3;
zh[t_] :=    1 + t  0;
h[t_] := {xh[t], yh[t], zh[t]};    h[t]

Solve[g[s] == h[t], {s, t}]

Schnittpunkt = h[0]

w[t_] := 
        Schnittpunkt + 
          t ((g[1] - g[0])/Norm[(g[1] - g[0])] + (h[1] - h[0])/
                  Norm[(h[1] - h[0])]);
w[t]

w[t Sqrt[5/2]/2]

{w[t Sqrt[5/2]/2]} // Transpose // MatrixForm

4

OA = {3, -2, 2};  OB = {-3, 5, 8};
OU = {2, 1, -3};  OV = {1, 5, 4};
OW = {6, -2, -1};
g[t_] := OA + t (OB - OA);
g[t]

Φ[λ_, μ_] := 
        OU + λ (OV - OU) + μ (OW - 
                OU);
Φ[λ, μ]

solv = 
      Solve[g[t]    == Φ[λ, μ], {t, λ, μ}] // Flatten

Schnittpunkt = g[t]    /.  solv

5

a

Φ[λ_, μ_] := {1, 
            2, -3} + λ {-1, 4, 7} + μ {4, -3, 
              2};
Φ[λ, μ]

Ψ[ν_, ξ_] := {5, -2, -3} + ν {22, -23, -4} + ξ {13, 0, 29};
Ψ[ν, ξ]

solv = 
      Solve[Ψ[ν, ξ]    == Φ[λ, μ], {ν, ξ, λ, μ}] // 
        Flatten

Kein Schnittpunkt: Ebenen parallel

b

Φ[λ_, μ_] := {5, 1, 
            8} + λ {1, 0, 3} + μ {2, 
              1, -1};
Φ[λ, μ]

Ψ[ν_, ξ_] := {11, 
            7, -16} + ν {6, 2, -1} + ξ {-1, -1, 
              3};
Ψ[ν, ξ]

solv = 
      Solve[Ψ[ν, ξ]    == Φ[λ, μ], {ν, ξ, λ, μ}] // 
        Flatten

(Φ[λ, μ] /. solv) // Simplify

(Ψ[ν, ξ] /. solv) // Simplify

Schnittgerade, hier dargestellt mit Parameter μ

(1 - t^6)/(1 - t) // Simplify

6

Remove["Global`*"]

Φ[λ_,μ_]:={2,3,1}+λ{4,-2,3}+μ{1,0,-2};
Φ[λ,μ]

solv=Solve[{x,y,z}==Φ[λ,μ],{λ,μ,x}]//Flatten

(x1-x==0 /.solv)

(%/.x1->x)//Simplify

7

Remove["Global`*"]

Φ[{x_,y_,z_}]:=3x-7z-21;

P1={0,0,z}; P2={0,y,0}; P3={x,0,0};

solv1=Solve[Φ[P1]==0,{z}]//Flatten

P1={0,0,z}/.solv1

solv2=Solve[Φ[P2]==0,{y}]//Flatten

P2={0,y,0}/.solv2

y ist beliebig. Setze y=1

y=1;

solv3=Solve[Φ[P3]==0,{x}]//Flatten

P3={x,0,0}/.solv3

Φ[λ_,μ_]:= P1+λ(P2-P1)+μ(P3-P1);
{Φ[λ,μ]}//Transpose//MatrixForm

8

Remove["Global`*"]

a

PA={4,3,-2}; PB={-3,1,2}; PC={1,0,2}; 
Φ[λ_,μ_]:= PA+λ(PB-PA)+μ(PC-PA);
{Φ[λ,μ]}//Transpose//MatrixForm

b

PA={2,-3,0}; PB={-4,6,2}; PC={0,0,9}; 
Φ[λ_,μ_]:= PA+λ(PB-PA)+μ(PC-PA);
{Φ[λ,μ]}//Transpose//MatrixForm

Lösungen Teil 2

1

Remove["Global`*"]

OA={-1,0,5}; OB={3,-4,7}; OC={2,2,3};
α = ArcCos[(OB-OA).(OC-OA)/(Norm[OB-OA] Norm[OC-OA])]

%/Degree//N

β = ArcCos[(OA-OB).(OC-OB)/(Norm[OA-OB] Norm[OC-OB])]

%/Degree//N

γ = ArcCos[(OA-OC).(OB-OC)/(Norm[OA-OC] Norm[OB-OC])]

%/Degree//N

Kontrolle

(α + β + γ)/Degree // N

2

Remove["Global`*"]

OP={1,2,0}; Φ[{x_,y_,z_}]:=x-y+2z-3; nVec={1,-1,2};
h[λ_]:=OP+λ nVec;
solv=Solve[Φ[h[λ]]==0,{λ}]//Flatten

OS=h[λ]/.solv

OPgespiegelt=OP+2(OS-OP)

d=Norm[2(OS-OP)]

Norm[OP-OPgespiegelt]

% //N

3

Remove["Global`*"]

OP={p1,p2,p3}; a={a1,a2,a3}; b={b1,b2,b3};
g1[λ_]:=OP+ λ a;
g2[μ_]:=OP+ μ b;

s1[λ_]:=OP+ λ (a/Norm[a]+b/Norm[b]);
s2[μ_]:=OP+ μ (a/Norm[a]-b/Norm[b]);

((a/Norm[a]+b/Norm[b]).(a/Norm[a]-b/Norm[b])//Simplify)/.{Abs[a1]^2->a1^2,Abs[a2]^2->a2^2,Abs[a3]^2->a3^2,Abs[b1]^2->b1^2,Abs[b2]^2->b2^2,Abs[b3]^2->b3^2}

%//Expand

4

Remove["Global`*"]

a={3,8,x}; b={3,-8,x};
a.b == Norm[a] Norm[b] Cos[60 Degree]

Solve[a.b == Norm[a] Norm[b] Cos[60 Degree],{x}]

%//N

5

Remove["Global`*"]

OA={2,-4,-9}; OB={0,6,1}; OX[s_]:={3,12,16}+s {3,10,11};
Simplify[(OA-OB).(OX[s]-OB)]==0

solv=Solve[(OA-OB).(OX[s]-OB)==0,{s}]//Flatten

OC=OX[s]/.solv

OD=OC+(OA-OB)

6

Remove["Global`*"]

HNFΦ[x_,y_,z_]:=(x+2y+3z-5)/Sqrt[1^2+2^2+3^2]; HNFΨ[x_,y_,z_]:=(x+2y+3z+2)/Sqrt[1^2+2^2+3^2];

HNFΦ[0,0,0]

HNFΨ[0,0,0]

HNFΨ[0,0,0]-HNFΦ[0,0,0]

N[%]

7

Remove["Global`*"]

NFFΦ[x_,y_,z_]:=(3x-y+3z-1)/Sqrt[3^2+(-1)^2+2^2]; HNFΨ[x_,y_,z_]:=(-6x+2y-4z-7)/Sqrt[(-6)^2+2^2+(-4)^2];  
Print[NFFΦ[x,y,z], "  ",HNFΨ[x,y,z]]

Normalenvektoren nicht paralle!

Solve[{NFFΦ[x,y,z]==0, HNFΨ[x,y,z]==0},{x,y,z}]

N[%]

Schnittgerade! Abstand = 0.

8

Remove["Global`*"]

OP={-1,0,3};
HNFΦ[x_,y_,z_]:=(-x+2y+5z+2)/Sqrt[(-1)^2+2^2+5^2]; HNFΦ[{x_,y_,z_}]:= HNFΦ[x,y,z];
HNFΦ[x,y,z]

HNFΦ[OP]

N[%]

9

Remove["Global`*"]

HNFΦ[x_,y_,z_]:=(3x-5y-4z-10)/Sqrt[3^2+(-5)^2+(-4)^2];
HNFΦ[x,y,z]

HNFΨ1[x_,y_,z_]:=(3x-5y-4z-10)/Sqrt[3^2+(-5)^2+(-4)^2]+4;
HNFΨ1[0,0,0]

N[%]

HNFΨ2[x_,y_,z_]:=(3x-5y-4z-10)/Sqrt[3^2+(-5)^2+(-4)^2]-4;
HNFΨ2[0,0,0]

N[%]

10

Remove["Global`*"]

HNFΦ[x_,y_,z_]:=(-x-2y+z-2)/Sqrt[(-1)^2+(-2)^2+(1)^2];
HNFΦ[x,y,z]

OA={3,4,2}; OB={1,-1,5}; OC={3,0,1}; nVec=Cross[(OB-OA),(OC-OA)]

HNFΨ[x_,y_,z_]:=({x,y,z}.nVec+dD)/Norm[nVec]; HNFΨ[{x_,y_,z_}]:=HNFΨ[x,y,z];
HNFΨ[x,y,z]

solv=Solve[HNFΨ[OA]==0,{dD}]//Flatten

HNFΨ[x_,y_,z_]:=({x,y,z}.nVec+dD)/Norm[nVec]/. solv;
HNFΨ[x,y,z]

Simplify[Sqrt[357] HNFΨ[x,y,z]-Sqrt[6] HNFΦ[x,y,z]]==0

Simplify[Sqrt[357] HNFΨ[x,y,z]+Sqrt[6] HNFΦ[x,y,z]]==0

11

Der Umkreismittelpunkt liegt auf den Senkrechten durch die Seitenmittelpunkte.

Remove["Global`*"]

OA={a1,a2,a3}; OB={b1,b2,b3}; OC={c1,c2,c3}; 
OM[λ_,μ_]:=OA+λ (OB-OA)+μ (OC-OA)//Simplify;
OM[λ,μ]

OM ist Lösung des Gleichungssystems:
r^2 = |OM-OA|^2, r^2 = |OM-Ob|^2,  r^2 = |OM-OC|^2
(3 Gelichungen, Unbekannte r, λ, μ)

{r^2==(OM[λ,μ]-OA), r^2==(OM[λ,μ]-OB), r^2==(OM[λ,μ]-OC)}

Solve[{r^2==(OM[λ,μ]-OA).(OM[λ,μ]-OA), r^2==(OM[λ,μ]-OB).(OM[λ,μ]-OB), r^2==(OM[λ,μ]-OC).(OM[λ,μ]-OC)},{λ,μ,r}]//Simplify

OMC=OA+(OB-OA)/2 //Simplify

OMA=OB+(OC-OB)/2 //Simplify

b

OA={-2,5,-5}; OB={3,-1,5}; OC={0,3,-1};
OM[λ_,μ_]:=OA+λ (OB-OA)+μ (OC-OA)//Simplify;
OM[λ,μ]

solv=Solve[{r^2==(OM[λ,μ]-OA).(OM[λ,μ]-OA), r^2==(OM[λ,μ]-OB).(OM[λ,μ]-OB), r^2==(OM[λ,μ]-OC).(OM[λ,μ]-OC)},{λ,μ,r}]//Simplify //Flatten

N[%]

OM[λ,μ]/.solv

N[%]

Lösungen Teil 3

1

Remove["Global`*"]

a={1,0,3}; b={2,-2,3};
Cross[a,b]

Norm[Cross[a,b]]

2

Remove["Global`*"]

OP={2,4,-5};
g[s_]:={1,0,-6}+s {-3,5,2};

(OP-g[s]).{-3,5,2}==0

solv=Solve[(OP-g[s]).{-3,5,2}==0,{s}]//Flatten

g[s]/.solv

N[%]

3

Remove["Global`*"]

OA={2,-1,5}; OB={3,3,0}; OC={-4,3,-2}; OD={-1,-3,2};
Volumen = 1/6 Det[{OC-OA, OB-OA, OD-OA}]

N[%]

Oberflaeche = 1/2 (Norm[Cross[OB-OA,OC-OA]]+ Norm[Cross[OC-OA,OD-OA]]+ Norm[Cross[OD-OA,OB-OA]]+  Norm[Cross[OC-OB,OD-OB]])

N[%]

4

Remove["Global`*"]

OA={0,5,0}; OB={5,0,6}; 
OC={5,13-4,0}; OD={0,13,6};

d=Volumen/Grundfläche

d = Det[{OB-OA,OC-OA,OD-OC}]

G = Cross[OB-OA,OD-OC]//Norm

N[%]

d/G

N[%]

5

Remove["Global`*"]

kF={30,-60,30}; a={1,2,0}; b={-2,0,2}; 
kF1[λ_,μ_]:= λ a+ μ b; kF2[ν_]:= ν Cross[a,b];
solv=Solve[kF==kF1[λ,μ]+kF2[ν],{λ,μ,ν}]//Flatten

kF1[λ,μ]/.solv

kF2[ν]/.solv

6

Remove["Global`*"]

OP={2,0,-3}; Φ[λ_,μ_]:={2,11,-16}+λ{-3,0,2}+μ{0,1,-3};
g[t_]:= OP + t Cross[{-3,0,2},{0,1,-3}]; g[t]

7

Remove["Global`*"]

OP={2,0,-1}; 
Φ[x_,y_,z_]:={x,y,z}.{2,-1,3}+4;
Ψ[x_,y_,z_]:={x,y,z}.{1,1,-2}-3;
Γ[λ_,μ_]:= OP + λ  {2,-1,3}+ μ {1,1,-2}; Γ[λ,μ]

{Γ[λ,μ]}//Transpose//MatrixForm

8

Remove["Global`*"]

a

OO={0,0,0}; OA={12,4,0}; OB={2,14,0}; h=10;
OM=(OA+OB)/2

nVec = h Cross[OA,OB]/Norm[Cross[OA,OB]]

OC=OM+nVec

b

ACB = ArcCos[(OA-OC).(OB-OC)/(Norm[OA-OC] Norm[OB-OC])]

ACB//N

ACB/Degree//N

c

nOAC=Cross[OA,OC]

nOBC=Cross[OB,OC]

α = 
      ArcCos[nOAC . nOBC / Norm[nOAC] / Norm[nOBC]]

α//N

αDeg=α/Degree//N

Winkel=180-αDeg

c

FlaecheDach= ( Norm[nOAC]+Norm[nOBC] )/2

FlaecheDach//N

d

Det[{OA,OB,OC}]/6

%//N