Lösungen

Remove["Global`*"]

1

k1 = {2, 5, 6};  k11 = 30 k1/Norm[k1]

% // N

k2 = {1, 5, 7}; k21 = 40 k2/Norm[k2]

% // N

k11 + k21

% // N

{k11,k21,k11+k21}//Transpose//MatrixForm

{k11,k21,k11+k21}//Transpose//N//MatrixForm

a={2,1,0}; b={0,2,1}; c={1,0,3};
solv = Solve[k11+k21==λ a+μ b+ν c,{λ, μ, ν}]//Flatten

% // N

λ a/.solv[[1]]

%//N

Norm[λ  a] /. solv[[1]]

% // N

1 Old

k1={2,5,6}; k1=k1/Norm[k1] 30;
k2={1,5,7}; k2=k2/Norm[k2] 40;
{k1,k2,k1+k2}//Transpose//MatrixForm

{k1,k2,k1+k2}//Transpose//N//MatrixForm

a={2,1,0}; b={0,2,1}; c={1,0,3};
solv=Solve[k1+k2==λ a+μ b+ν c,{λ, μ, ν}]//Flatten

λ=λ/.solv

%//N

a1=λ a

%//N

Norm[a1]

%//N

2

kugel[x_,y_,z_]:= ((x-2)^2+(y-3)^2+(z-3)^2==36)

kugel[3,y,0]

solv=Solve[kugel[3,y,0],{y}]//Flatten

%//N

P1={3,y,0}/.solv[[1]]

P2={3,y,0}/.solv[[2]]

M={2,3,3}

d=Norm[Cross[P2-P1,M-P1]]/Norm[M-P1]

%//N

d=Norm[Cross[P2-P1,M-P2]]/Norm[M-P2]

3

Ebene (P1, P2, P3), Q1 spiegeln => Schnittpunkt L,, Q2, Volumen (Q1,P1,P2,P3,Q2) = ?

P1={0,0,3};
P2={0,6,0};
P3={5,0,0};
Q1={2,4,0};

Winkel zwischen P1Q1 und P1Q2 ?

a

Clear[λ]

P1={0,0,3};P2={0,6,0};P3={5,0,0};Q1={2,4,0};
g[λ_]:=Q1+λ Cross[P2-P1,P3-P1];
e[μ_, ν_]:=P1+μ (P2-P1)+ ν (P3-P1);
solv=Solve[g[λ]==e[μ, ν],{λ, μ, ν }]//Flatten

Lot=g[λ]/.solv[[1]]

%//N

b

Q2=Q1+2(Lot-Q1)

%//N

c

V=1/6(Abs[Det[{P1-Q1,P2-Q1,P3-Q1}]] + Abs[Det[{P1-Q2,P2-Q2,P3-Q2}]])

d

Norm[Cross[P2-P1,P3-P1]]/2

%//N

e

ArcCos[(Q2-P1).(Q1-P1)/(Norm[Q2-P1] Norm[Q1-P1])]

%//N

%/Degree

4

a

Clear[f,f1]

f[x_,y_]:=Sin[x +y];

Plot3D[Sin[x +y],{x,0,Pi},{y,0,Pi}];

b

f1[x_,y_]=Sin[x +y];

tgEbene[λ_, μ_]:= {Pi/2,Pi/2,f[x,y]}+λ  D[{x,y,f1[x,y]},x]+  μ D[{x,y,f1[x,y]},y]; tgEbene[λ, μ]

tgEbene[-Pi/2, -Pi/2]/.{x->Pi/2,y->Pi/2}

%//N

c

Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]]

Sqrt[Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]].Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]]]

Sqrt[Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]].Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]]][[1]][[2]][[2]][[1]][[1]][[1]]

FullForm[Sqrt[Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]].Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]]]/.{y->1,x->2}]

inhalt=NIntegrate[Evaluate[Sqrt[Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]].Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]]]/.{x->x1,y->y1}],{x1,0,Pi},{y1,0,Pi}]

inhalt=NIntegrate[Evaluate[Sqrt[Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]].Cross[D[{x,y,f1[x,y]},x],D[{x,y,f1[x,y]},y]]]],{x,0,Pi},{y,0,Pi}]

d

Sqrt[D[{x,y,f1[x,y]},y].D[{x,y,f1[x,y]},y]]

Laenge1= NIntegrate[Evaluate[Sqrt[D[{x,y,f1[x,y]},y].D[{x,y,f1[x,y]},y]]/.{x->Pi,y->y1}],{y1,0,Pi}]

Laenge2= NIntegrate[Evaluate[Sqrt[D[{x,y,f1[x,y]},y].D[{x,y,f1[x,y]},y]]/.{x->0,y->y1}],{y1,0,Pi}]

Laenge3= NIntegrate[Evaluate[Sqrt[D[{x,y,f1[x,y]},x].D[{x,y,f1[x,y]},x]]/.{y->Pi,x->x1}],{x1,0,Pi}]

Laenge4= NIntegrate[Evaluate[Sqrt[D[{x,y,f1[x,y]},x].D[{x,y,f1[x,y]},x]]/.{y->0,x->x1}],{x1,0,Pi}]

Laenge1+Laenge2+Laenge3+Laenge4