Lösungen

1

Remove["Global`*"]

A={{2,-2},{-3,1}}; A//MatrixForm

B={{1,3},{-1,2}};B//MatrixForm

v={{4,-3}}//Transpose

A.v

v={4,-3}

A.v

a

2B-A//MatrixForm

x1=(2B-A).v

b

2B+A//MatrixForm

(2B+A).{x11,x21}==v

((2B+A).{x11,x21}==v)//Solve

%//N

Eindeutig

2

Remove["Global`*"]

OA1={-1,1,1}; a={2,1,1}; b={4,5,-2};
OA2={1,-1,2}; c= {1,2,-1}; d={-3,2,-4};
v1[λ_,μ_]:={-1,1,1}+λ {2,1,1}+μ {4,5,-2};
v2[υ_,σ_]:={1,-1,2}+υ {1,2,-1}+σ {-3,2,-4};
Q={-5,-6,8}; oO={0,0,0}; OQ=Q-oO;

a

n1=Cross[a,b]

n2=Cross[c,d]

α=ArcCos[n1.n2/(Norm[n1] Norm[n2])]

%//N

%/(2Pi) 360

b

gL1[t_]:=OQ+t n1;
solv1=Solve[v1[λ,μ]==gL1[t],{t,λ,μ}]//Flatten

OL=gL1[t]/.solv1

N[%]

c

HNF2[x_,y_,z_,dd_]=n2.{x,y,z}/Norm [n2]+dd; (HNF2[x,y,z,dd])==0/.{x-> OA2[[1]] ,y-> OA2[[2]],z-> OA2[[3]]}

solv2=Solve[%,{dd}]//Flatten;  dd=dd/.solv2

HNF2[{x_,y_,z_}]:=HNF2[x,y,z,dd];  HNF2[OQ]

N[%]

d

solv3[ σ_]:=Solve[v1[λ,μ]==v2[υ,σ],{λ,μ,υ}]//Flatten; solv3[ σ]

((v2[υ,σ])/. solv3[ σ])//Simplify

((v1[λ,μ])/. solv3[ σ])//Simplify

gS[ σ_]:=((v2[υ, σ])/. solv3[ σ])//Simplify;
gS[ sigma]

aS=gS[ 1]-gS[0]

Flaeche=Norm[Cross[gS[ 1]-gS[0],OQ-gS[0] ]]

h=Flaeche/Norm[aS]

N[%]

3

Remove["Global`*"]

a1T[r_]:={r,-3,-2};  a2T={-1,1,-1}; a3T={3,3,0};

A[r_]:=Transpose[{a1T[r], a2T, a3T}]; A[r]//MatrixForm

a

Det[A[r]]

b

Solve[Det[A[r]]==0]

c

r ungleich - 7

d

Solve[Det[A[r]]==1]

%//N

4

Remove["Global`*"]

a1T[r_]:={r,-3,-2};  a2T={-1,1,-1}; a3T={3,3,0};

A[r_]:=Transpose[{a1T[r], a2T, a3T}]; A[r]//MatrixForm

a

Vwuerfel=s^3

Voktaeder=8 1/6 (s/2)^3

Vwuerfel/Voktaeder

b

E1={0,1,1}; E2={1,0,1}; E3={1,1,0}; 
E4={2,1,1}; E5={1,2,1}; E6={1,1,2};

V=Det[{E2-E1,E3-E1,E4-E1}]

V=Det[{E2-E1,E3-E1,E5-E1}]

Fl=Norm[Cross[E2-E1,E3-E1]]

V/Fl

N[%]

c

n1=Cross[E3-E1,E2-E1]

n2=Cross[E6-E1,E2-E1]

α=ArcCos[n1.n2/(Norm[n1] Norm[n2])]

%//N

Pi-%

%/(2Pi) 360

5

Remove["Global`*"]

Darstellung nur Dreiecke

P0={0,0}; P1={5,0}; P2={4,1}; P3={3.5,2}; P4={2,6}; P5={-1,5};
Show[Graphics[Line[{P0,P1,P2,P0,P2,P3,P0,P3,P4,P0,P4,P5,P0}]]]

Nur Dreiecke statt Parallelogramme gezeigt.

a

Fl[A_,B_]:=A[[1]] B[[2]]-A[[2]] B[[1]];

Fl[P1,P2]

Fl[P2,P3]

Fl[P3,P4]

Fl[P4,P5]

Fl[P1,P2]+Fl[P2,P3]+Fl[P3,P4]+Fl[P4,P5]

b

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

MatrixForm[MD[34.7 Degree]]

c

P6=MD[34.7 Degree].P5

6

Remove["Global`*"]

z1=1-2I; z2=-2-3I; z3=-1-5I;

a

w1=5 z1^2-z2/z3

w1//N

b

Solve[z^5==z1,{z}]

M=Solve[z^5==z1,{z}]//N//Flatten

c

zz[k_]:=z/.M[[k]];
zz[1]

Sum[zz[k],{k,1,5}]//Chop

7

Remove["Global`*"]

PA={2,1,0}; PB={3,8,0}; PC={-1,-3,5}; PD={6,-9,4};

Show[Graphics3D[Line[{PA,PB,PC,PA,PD,PB,PC,PD}]]]

Det[{PB-PA,PC-PA,PD-PA}]

Det[{PB-PA,PC-PA,PD-PA}]==0

==> Tetraeder

Fl[a_,b_]:=Norm[1/2 Cross[a,b]]//N

Fl[PB-PA,PC-PA]

Fl[PD-PA,PC-PA]

Fl[PD-PA,PB-PA]

Fl[PB-PD,PC-PD]

Fl[PB-PA,PC-PA]+Fl[PD-PA,PC-PA]+Fl[PD-PA,PB-PA]+Fl[PB-PD,PC-PD]

8

Remove["Global`*"]

M={2,1,5}; r=4; T[z_]:={1,-0.5,z}

(M- T[z]).(M- T[z])==r^2

solv=Solve[(M- T[z]).(M- T[z])==r^2,{z}]//Flatten

T[z]= T[z]/.solv[[1]]

((M-{x,y,z}).(M- T[z])==r^2)//Simplify

(((M-{x,y,z}).(M- T[z])==r^2)/.{y->0,z->0})//Simplify