Lösungen

1

Remove["Global`*"]

M={6,5};
Pk[x_,y_]:={x,y};
r=4;
Kr[x_,y_]:=(Pk[x,y]-M).(Pk[x,y]-M)-r^2;
Kr[{x_,y_}]:=Kr[x,y];
P={-1,0};

b

d1=Norm[M-P]-r

%//N

d2=Norm[M-P]+r

%//N

{Kr[x,y]==0,(Pk[x,y]-P).(Pk[x,y]-P)==d1  d2}//Simplify

solv=Solve[{Kr[x,y]==0,(Pk[x,y]-P).(Pk[x,y]-P)==d1  d2},{x,y}]

T1={x,y}/.solv[[2]]

N[%]

T2={x,y}/.solv[[1]]

N[%]

S1=P+(M-P)/Norm[M-P] d1

N[%]

S2=P+(M-P)/Norm[M-P] d2

N[%]

Q=T1+(T2-T1)/3   2

N[%]

c

gPQ[t_]:=P+t (Q-P);

{Kr[x,y]==0,Pk[x,y]==gPQ[t]}

solv2=Solve[{Kr[x,y]==0,Pk[x,y]==gPQ[t]},{x,y,t}]

N[%]

H1={x,y}/.solv2[[1]]

N[%]

H2={x,y}/.solv2[[2]]

N[%]

a

Show[Graphics[{
Circle[M,r],
PointSize[0.03],Point[M],Point[P],Point[T1],Point[T2],
Point[P+(M-P)/Norm[M-P] d2],Point[S1],
Point[Q],Point[H1],Point[H2],
Line[{M,T1,P,M,T2,P}],Line[{P,S2}],
Line[{T1,T2}],Line[{P,H2}]

}]]

d

Norm[T1-P]^2

N[%]

Norm[T2-P]^2

N[%]

Norm[H1-P] Norm[H2-P]

N[%]

2

Remove["Global`*"]

a

B={{r,2},{3,2}}; B//MatrixForm

Inverse[B]//MatrixForm

%//Simplify//MatrixForm

b

A[r_]:={{r,2,3r},{3,2,1},{1,3,-2}}; A[r]//MatrixForm

Det[A[r]]

c

Inverse[A[r]]//MatrixForm

Solve[Det[A[r]]==0,{r}]

d

A (1) X Inv[A (1)] + E = A (0) -  A (r) X Inv[A (1)]
A (1) X Inv[A (1)] + A (r) X Inv[A (1)] = A (0) - E
(A (1) + A (r))  X Inv[A (1)] = (A (0) - E)   | von rechts mal A (1)
von links mal Inv[(A (1) + A (r))]     |     (A (1) + A (r))  X E = (A (0) - E) A (1)
 E X = Inv[(A (1) + A (r))] (A (0) - E) A (1)
Resultat :
  X = Inv[(A (1) + A (r))] (A (0) - E) A (1)
=== === === === === === === === === === ==

X[x11_,x12_,x13_,x21_,x22_,x23_,x31_,x32_,x33_]:={{x11,x12,x13},{x21,x22,x23},{x31,x32,x33}};
X[x11,x12,x13,x21,x22,x23,x31,x32,x33]//MatrixForm

A[1].X[x11,x12,x13,x21,x22,x23,x31,x32,x33].Inverse[A[1]]+IdentityMatrix[3]==A[0]-A[r].X[x11,x12,x13,x21,x22,x23,x31,x32,x33].Inverse[A[1]]

solv3=Solve[A[1].X[x11,x12,x13,x21,x22,x23,x31,x32,x33].Inverse[A[1]]+IdentityMatrix[3]==A[0]-A[r].X[x11,x12,x13,x21,x22,x23,x31,x32,x33].Inverse[A[1]],{ x11,x12,x13,x21,x22,x23,x31,x32,x33}]//Flatten

Xres[r_]:=X[x11,x12,x13,x21,x22,x23,x31,x32,x33]/.solv3;
Xres[r]//MatrixForm

e

(Xres[r]/.r->2)//MatrixForm

N[%]

{{0.55`,2.65`,0.9`},{0.95`,-1.15`,1.1`},{-0.05`,-0.15`,-0.9`}}//MatrixForm

(Xres[r]/.r->-3)//MatrixForm

Existiert nicht!

3

Vorbemerkung: Es gibt 2 reelle Lösungen. Der Rechte Winkel kann bei A, B oder C liegen. Im Falle dass er bei A liegt, gibt es keine reelle Lösung.

3.1  Rechter Winkel bei B

Remove["Global`*"]

a

pA[y_]:={16,y,0}; pB={13,7,12}; pC={2,2,8};
{pA[y],pB,pC}

solve4=Solve[(pB-pA[y]).(pC-pB) == 0,{y}]//Flatten

pA1=pA[y]/.solve4

b

{Norm[pB-pA1],Norm[pC-pB],Norm[pC-pA1]}

Gleichschenklig

c

pD=pA1+(pC-pB)

{Norm[pA1-pB],Norm[pC-pD]}

{Norm[pA1-pD],Norm[pC-pB]}

Show[Graphics3D[{PointSize[0.03],Point[pA1],Point[pB],Point[pC],Point[pD],Line[{pA1,pB,pC,pD,pA1}]}],AspectRatio->Automatic]

d

gS[t_]:=pA1+1/2(pC-pA1) + t Cross[pB-pA1,pD-pA1];
gS[t]

Vol[t_]:=Det[{pB-pA1,pD-pA1,gS[t]-pA1}]/3;
Vol[t]

solv5=Solve[Sqrt[Vol[t]^2]==1944,{t}]//Flatten

N[%]

pS1=gS[t]/.solv5[[1]]

pS2=gS[t]/.solv5[[2]]

Show[Graphics3D[{PointSize[0.03],Point[pA1],Point[pB],Point[pC],Point[pD],Point[pS2],Line[{pA1,pB,pS2,pB,pC,pS2,pC,pD,pS2,pD,pA1,pS2}]}],AspectRatio->Automatic]

(* Kontrolle *)     Det[{pB-pA1,pD-pA1,pS2-pA1}]/3

3.2  Rechter Winkel bei C (Variante problematisch)

(* Variante problematisch, da ABCD Rechteck! *)  Remove["Global`*"]

a

pA[y_]:={16,y,0}; pB={13,7,12}; pC={2,2,8};
{pA[y],pB,pC}

solve4=Solve[(pA[y]-pC).(pB-pC) == 0,{y}]//Flatten

N[%]

pA1=pA[y]/.solve4

N[%]

b

{Norm[pB-pA1],Norm[pC-pB],Norm[pC-pA1]}

Nicht gleichschenklig

c

pD=pA1+(-1)(pC-pB)

N[%]

{Norm[pA1-pB],Norm[pC-pD]}

{Norm[pA1-pD],Norm[pC-pB]}

Show[Graphics3D[{PointSize[0.03],Point[pA1],Point[pB],Point[pC],Point[pD],Line[{pA1,pC,pB,pD,pA1}]}],AspectRatio->Automatic]

d

gS[t_]:=pA1+1/2(pB-pA1) + t Cross[pC-pA1,pD-pA1];
gS[t]

Vol[t_]:=Det[{pC-pA1,pD-pA1,gS[t]-pA1}]/3;
Vol[t]

solv5=Solve[Sqrt[Vol[t]^2]==1944,{t}]//Flatten

N[%]

pS1=gS[t]/.solv5[[1]]

N[%]

pS2=gS[t]/.solv5[[2]]

N[%]

Show[Graphics3D[{PointSize[0.03],Point[pA1],Point[pB],Point[pC],Point[pD],Point[pS2],Line[{pA1,pC,pS2,pC,pB,pS2,pB,pD,pS2,pD,pA1,pS2}]}],AspectRatio->Automatic]

(* Kontrolle *)     Det[{pC-pA1,pD-pA1,pS2-pA1}]/3

3.3  Rechter Winkel bei A: Keine reelle Lösung!

Remove["Global`*"]

a

pA[y_]:={16,y,0}; pB={13,7,12}; pC={2,2,8};
{pA[y],pB,pC}

solve4=Solve[(pC-pA[y]).(pB-pA[y]) == 0,{y}]//Flatten

N[%]

pA1=pA[y]/.solve4

N[%]

Keine reelle Lösung.

4

Remove["Global`*"]

P0={0,0};P1={5,0};P2={4,1};P3={3.5,2};P4={2,6};P5={-1,5};
A[p1_,p2_]:=p1[[1]] p2[[2]]- p2[[1]] p1[[2]];

a

A[P1,P2]/2 //N

A[P2,P3]/2

A[P3,P4]/2

A[P4,P5]/2

(A[P1,P2]+A[P2,P3]+A[P3,P4]+A[P4,P5])/2

b

Dr[ϕ_]:={{Cos[ϕ],-Sin[ϕ]},{Sin[ϕ],Cos[ϕ]}} ; Dr[ϕ]//MatrixForm

Dr[34.7 Degree]//MatrixForm

c

P6=Dr[34.7 Degree].P5

Show[Graphics[{PointSize[0.03],Point[P0],Point[P1],Point[P2],Point[P3],Point[P4],Point[P5],Point[P6],
Line[{P0,P1,P2,P0,P2,P3,P0,P3,P4,P0,P4,P5,P0,P5,P6,P0}]}]]

5

Remove["Global`*"]

pA={2,1,0}; pB={3,8,0}; pC={-1,-3,5}; pD={6,-9,4};

a

Show[Graphics3D[{PointSize[0.03],Point[pA],Point[pB],Point[pC],Point[pD],
Line[{pA,pB,pC,pA,pD,pB,pD,pC}]}]]

Vol=Det[{pB-pA,pC-pA,pD-pA}]/6

a

InhFlaeche=1/2 (Norm[Cross[pB-pA,pC-pA]]+Norm[Cross[pA-pD,pB-pD]]+Norm[Cross[pB-pD,pC-pD]]+Norm[Cross[pC-pD,pA-pD]])

N[%]

5

Remove["Global`*"]

M1={{-11,8},{8,1}}; M1//MatrixForm

M2={{2,-6},{-6,-7}}; M2//MatrixForm

a

Eigenvalues[M1]

b

Eigenvectors[M1]

Eigenvectors[M1][[1]]//Norm

nEV1[k_]:=Eigenvectors[M1][[k]]/Norm[Eigenvectors[M1][[k]]]//N

nEV1[1]

nEV1[2]

c

Eigenvalues[M2]

d

nEV2[k_]:=Eigenvectors[M2][[k]]/Norm[Eigenvectors[M2][[k]]]//N

nEV2[1]

nEV2[2]

e

Gleiche Eigenvektoren, verschiedene Eigenwerte

f

M1.M2//MatrixForm

M2.M1//MatrixForm

Kommutativ

g

Eigenvalues[M1.M2]

{5,-10} {-15,5}

Produkte der EW von M1 und M2

h

nEV3[k_]:=Eigenvectors[M1.M2][[k]]/Norm[Eigenvectors[M1.M2][[k]]]//N

nEV3[1]

nEV3[2]

EV von M1 und M2

i

Eigenvalues[M1.M2.M1]

{-15,5}{5,-10}{-15,5}

Produkte der EW von M1 und M2 und M1

j

a={2,4}; b={3,2}; o={0,0};

FlProd[p1_,p2_]:=p1[[1]] p2[[2]]- p2[[1]] p1[[2]];

a1=M1.a

b1=M1.b

FlProd[a,b]

FlProd[a1,b1]

Det[M1]

Eigenvalues[M1][[1]] Eigenvalues[M1][[2]]

FlProd[a,b] Det[M1] ==FlProd[a1,b1]