Lösungen Teil 1

(1)

Lösungen Teil 1

1

OA=83, 0, 4<; OB=81, 1, 1<; OC=8-7, 5, 11<; Solve@HOB-OALŠ lHOC-OAL,8l<D

8<

Die Punkte liegen icht auf einer Geraden.

2

Ÿ a

a1=81, 3, 0<; a2=86, 2, 0<; r1=82,-3, 1<; r2=8-1,-4, 1<; Solve@a1Š la2,8l<D

8<

Richtungsvektoren nicht parallel

solv=Solve@r1+ la1Šr2+ ma2,8l,m<D ••Flatten 9l ®0,m ® 1

€€€€2=

Schnittpunkt=r1+ la1•. solv@@1DD 82,-3, 1<

Ÿ b

a1=84, 3, 0<; a2=8-8,-6, 0<; r1=80, 0, 0<; r2=810, 6, 3<; Solve@a1Š la2,8l<D

99l ® -1

€€€€2==

Richtungsvektoren parallel

solv=Solve@r1+ la1Šr2+ ma2,8l,m<D ••Flatten 8<

(2)

Kein Schnittpunkt

Ÿ c

a1=80, 0, 1<; a2=82, 0,-1<; r1=83, 0, 5<; r2=81,-3, 6<; Solve@a1Š la2,8l<D

8<

Richtungsvektoren nicht parallel

solv=Solve@r1+ la1Šr2+ ma2,8l,m<D ••Flatten 8<

Kein Schnittpunkt

Ÿ d

a1=87, 2,-1<; a2=8-14,-4, 2<; r1=84,-3, 2<; r2=8-10,-7, 4<; Solve@a1Š la2,8l<D

99l ® -1

€€€€2==

Richtungsvektoren parallel

solv=Solve@r1+ la1Šr2+ ma2,8l,m<D ••Flatten 8l ® -2-2m<

3

xg@s_D:= -1+s 6;

yg@s_D:= -4+s 2;

zg@s_D:= 1+s 0;

g@s_D:=8xg@sD, yg@sD, zg@sD<; g@sD 8-1+6 s,-4+2 s, 1<

xh@t_D:= 2+t 1;

yh@t_D:= -3+t 3;

zh@t_D:= 1+t 0;

h@t_D:=8xh@tD, yh@tD, zh@tD<; h@tD 82+t,-3+3 t, 1<

Solve@g@sDŠh@tD,8s, t<D 99s® 1

€€€€2, t®0==

(3)

Schnittpunkt=h@0D 82,-3, 1<

w@t_D:=Schnittpunkt+

t HHg@1D-g@0DL •Norm@Hg@1D-g@0DLD+Hh@1D-h@0DL •Norm@Hh@1D-h@0DLDL; w@

tD

92+2$%%%%%%%2

€€€€5 t,-3+2$%%%%%%%2

€€€€5 t, 1=

w@t Sqrt@5•2D •2D 82+t,-3+t, 1<

8w@t Sqrt@5•2D •2D< ••Transpose••MatrixForm i

kjjjjj jj

2+t -3+t 1

y {zzzzz zz

4

OA=83,-2, 2<; OB=8-3, 5, 8<; OU=82, 1,-3<; OV=81, 5, 4<; OW=86,-2,-1<;

g@t_D:=OA+t HOB-OAL; g@tD

83-6 t,-2+7 t, 2+6 t<

F@l_,m_D:=OU+ l HOV-OUL+ m HOW-OUL; F@l,mD

82- l +4m, 1+4l -3m,-3+7l +2m<

solv=Solve@g@tD Š F@l,mD,8t,l,m<D ••Flatten 8t® -3,l ® -3,m ®4<

Schnittpunkt=g@tD •. solv 821,-23,-16<

5

Ÿ a

F@l_,m_D:=81, 2,-3<+ l 8-1, 4, 7<+ m 84,-3, 2<; F@l,mD

81- l +4m, 2+4l -3m,-3+7l +2m<

(4)

Y@n_,x_D:=85,-2,-3<+ n 822,-23,-4<+ x 813, 0, 29<; Y@n,xD

85+22n +13x,-2-23n,-3-4n +29x<

solv=Solve@Y@n,xD Š F@l,mD,8n,x,l,m<D ••Flatten 8<

Kein Schnittpunkt: Ebenen parallel

Ÿ b

F@l_,m_D:=85, 1, 8<+ l 81, 0, 3<+ m 82, 1,-1<; F@l,mD

85+ l +2m, 1+ m, 8+3l - m<

Y@n_,x_D:=811, 7,-16<+ n 86, 2,-1<+ x 8-1,-1, 3<; Y@n,xD

811+6n - x, 7+2n - x,-16- n +3x<

solv=Solve@Y@n,xD Š F@l,mD,8n,x,l,m<D ••Flatten 9n ® -6

€€€€7 + €€€€m

7,x ® 30

€€€€€€€

7 - €€€€€€€€€5m

7 ,l ® -24

€€€€€€€

7 - €€€€€€€€€3m 7 = HF@l,mD •. solvL ••Simplify

9€€€€€€€€€€€€€€€€€€€€€€€€€11H1+ mL

7 , 1+ m,-16

€€€€€€€

7 H1+ mL=

HY@n,xD •. solvL ••Simplify 9€€€€€€€€€€€€€€€€€€€€€€€€€11H1+ mL

7 , 1+ m,-16

€€€€€€€

7 H1+ mL=

Schnittgerade, hier dargestellt mit Parameter m

H1-t ^ 6L • H1-tL ••Simplify 1+t+t²+t³+t⁴+t⁵

6

Remove["Global`*"]

F[l_,m_]:={2,3,1}+l{4,-2,3}+m{1,0,-2};

F[l,m]

82+4l + m, 3-2l, 1+3l -2m<

solv=Solve[{x,y,z}==F[l,m],{l,m,x}]//Flatten 9l ® 3-y

€€€€€€€€€€€€

2 ,m ® 1

€€€€4 H11-3 y-2 zL, x® 1

€€€€4 H43-11 y-2 zL=

(5)

(x1-x==0 /.solv) x1+ 1

€€€€4 H-43+11 y+2 zLŠ0

(%/.x1->x)//Simplify 4 x+11 y+2 zŠ43

7

F[{x_,y_,z_}]:=3x-7z-21;

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

solv1=Solve[F[P1]==0,{z}]//Flatten 8z® -3<

P1={0,0,z}/.solv1 80, 0,-3<

solv2=Solve[F[P2]==0,{y}]//Flatten 8<

P2={0,y,0}/.solv2 80, y, 0<

y ist beliebig. Setze y=1

y=1;

solv3=Solve[F[P3]==0,{x}]//Flatten 8x®7<

P3={x,0,0}/.solv3 87, 0, 0<

F[l_,m_]:= P1+l(P2-P1)+m(P3-P1);

{F[l,m]}//Transpose//MatrixForm i

kjjjjj jj

7m l

-3+3l +3m y {zzzzz zz

8

(6)

Ÿ a

PA={4,3,-2}; PB={-3,1,2}; PC={1,0,2};

F[l_,m_]:= PA+l(PB-PA)+m(PC-PA);

kjjjjj jj

4-7l -3m 3-2l -3m -2+4l +4m

y {zzzzz zz

Ÿ b

PA={2,-3,0}; PB={-4,6,2}; PC={0,0,9};

F[l_,m_]:= PA+l(PB-PA)+m(PC-PA);

kjjjjj jj

2-6l -2m -3+9l +3m 2l +9m

y {zzzzz zz

Lösungen Teil 2

1

OA={-1,0,5}; OB={3,-4,7}; OC={2,2,3};

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

€€€€p 2

%/Degree//N 90.

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

ArcCosA 6

€€€€€€€€€€€€€•!!!!!!!53 E

%/Degree//N 34.4962

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

ArcCosA$%%%%%%%%%%€€€€€€€17 53 E

(7)

%/Degree//N 55.5038

Ÿ Kontrolle

Ha + b + gL •Degree••N 180.

2

OP={1,2,0}; F[{x_,y_,z_}]:=x-y+2z-3; nVec={1,-1,2};

h[l_]:=OP+l nVec;

solv=Solve[F[h[l]]==0,{l}]//Flatten 9l ® 2

€€€€3=

OS=h[l]/.solv 95

€€€€3, 4

€€€€3=

OPgespiegelt=OP+2(OS-OP) 97

€€€€3, 2

€€€€3, 8

€€€€3=

d=Norm[2(OS-OP)]

4$%%%%%%%2

€€€€3

Norm[OP-OPgespiegelt]

4$%%%%%%%2

€€€€3

% //N 3.26599

3

OP={p1,p2,p3}; a={a1,a2,a3}; b={b1,b2,b3};

g1[l_]:=OP+ l a;

g2[m_]:=OP+ m b;

s1[l_]:=OP+ l (a/Norm[a]+b/Norm[b]);

s2[m_]:=OP+ m (a/Norm[a]-b/Norm[b]);

(8)

((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}

Ha1²b1²+a2²b1²+a1²b2²+a2²b2²+a1²b3²+a2²b3²-

a1²Hb1²+b2²+b3²L-a2²Hb1²+b2²+b3²LL • HHa1²+a2²+a3²L Hb1²+b2²+b3²LL

%//Expand 0

4

a={3,8,x}; b={3,-8,x};

a.b == Norm[a] Norm[b] Cos[60 Degree]

-55+x²Š 1

€€€€2 H73+Abs@xD²L

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

99x® -•!!!!!!!!!!183=,9x®•!!!!!!!!!!183==

%//N

88x® -13.5277<,8x®13.5277<<

5

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

Simplify[(OA-OB).(OX[s]-OB)]==0 -204H1+sLŠ0

solv=Solve[(OA-OB).(OX[s]-OB)==0,{s}]//Flatten 8s® -1<

OC=OX[s]/.solv 80, 2, 5<

OD=OC+(OA-OB) 82,-8,-5<

6

(9)

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

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

HNFF[0,0,0]

- 5

€€€€€€€€€€€€€•!!!!!!!14

HNFY[0,0,0]

$%%%%%%%2

€€€€7

HNFY[0,0,0]-HNFF[0,0,0]

$%%%%%%%2

€€€€7 + 5

€€€€€€€€€€€€€•!!!!!!!14

N[%]

1.87083

7

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

HNFY[x_,y_,z_]:=(-6x+2y-4z-7)/Sqrt[(-6)^2+2^2+(-4)^2];

Print[NFFF[x,y,z], " ",HNFY[x,y,z]]

-1+3 x-y+3 z

€€€€€€€€€€€€€€€€•!!!!!!!€€€€€€€€€€€€€€€€14 €€€€€€€€€

-7-6 x+2 y-4 z

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

2•!!!!!!!14

Normalenvektoren nicht paralle!

Solve[{NFFF[x,y,z]==0, HNFY[x,y,z]==0},{x,y,z}]

99x® -25

€€€€€€€

6 + y

€€€€3, z® 9

€€€€2==

N[%]

88x® -4.16667+0.333333 y, z®4.5<<

Schnittgerade! Abstand = 0.

8

(10)

OP={-1,0,3};

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

HNFF[x,y,z]

2-x+2 y+5 z

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€•!!!!!!!30

HNFF[OP]

3$%%%%%%%6

€€€€5

N[%]

3.28634

9

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

HNFF[x,y,z]

-10+3 x-5 y-4 z

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

5•!!!!2

HNFY1[x_,y_,z_]:=(3x-5y-4z-10)/Sqrt[3^2+(-5)^2+(-4)^2]+4;

HNFY1[0,0,0]

4-•!!!!2 N[%]

2.58579

HNFY2[x_,y_,z_]:=(3x-5y-4z-10)/Sqrt[3^2+(-5)^2+(-4)^2]-4;

HNFY2[0,0,0]

-4-•!!!!2 N[%]

-5.41421

10

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

HNFF[x,y,z]

-2-x-2 y+z

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€•!!!!6

(11)

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

817,-2, 8<

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

HNFY[x,y,z]

dD+17 x-2 y+8 z

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€•!!!!!!!!!!357 €€€€€€€€€€€€€€€

solv=Solve[HNFY[OA]==0,{dD}]//Flatten 8dD® -59<

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

HNFY[x,y,z]

-59+17 x-2 y+8 z

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€•!!!!!!!!!!357€€€€€€€€€€€€€€€€€€

Simplify[Sqrt[357] HNFY[x,y,z]-Sqrt[6] HNFF[x,y,z]]==0 -57+18 x+7 zŠ0

Simplify[Sqrt[357] HNFY[x,y,z]+Sqrt[6] HNFF[x,y,z]]==0 -61+16 x-4 y+9 zŠ0

11

Ÿ Der Umkreismittelpunkt liegt auf den Senkrechten durch die Seitenmittelpunkte.

OA={a1,a2,a3}; OB={b1,b2,b3}; OC={c1,c2,c3};

OM[l_,m_]:=OA+l (OB-OA)+m (OC-OA)//Simplify;

OM[l,m]

8b1l +c1m -a1H-1+ l + mL, b2l +c2m -a2H-1+ l + mL, b3l +c3m -a3H-1+ l + mL<

OM ist Lösung des Gleichungssystems:

r^2 = |OM-OA|^2, r^2 = |OM-Ob|^2, r^2 = |OM-OC|^2 (3 Gelichungen, Unbekannte r, l, m)

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

8r²Š8-a1+b1l +c1m -a1H-1+ l + mL,

-a2+b2l +c2m -a2H-1+ l + mL,-a3+b3l +c3m -a3H-1+ l + mL<, r²Š8-b1+b1l +c1m -a1H-1+ l + mL,-b2+b2l +c2m -a2H-1+ l + mL,

-b3+b3l +c3m -a3H-1+ l + mL<, r²Š8-c1+b1l +c1m -a1H-1+ l + mL, -c2+b2l +c2m -a2H-1+ l + mL,-c3+b3l +c3m -a3H-1+ l + mL<<

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

(12)

99r® -1

€€€€2 ,HHHa1²+a2²+a3²-2 a1 b1+b1²-2 a2 b2+b2²-2 a3 b3+b3²L Ha1²+a2²+a3²-2 a1 c1+c1²-2 a2 c2+c2²-2 a3 c3+c3²L Hb1²+b2²+b3²-2 b1 c1+c1²-2 b2 c2+c2²-2 b3 c3+c3²LL •

Ha1²b2²+a1²b3²-2 a1 b2²c1-2 a1 b3²c1+b2²c1²+b3²c1²-2 a1²b2 c2+ 2 a1 b1 b2 c2+2 a1 b2 c1 c2-2 b1 b2 c1 c2+a1²c2²-2 a1 b1 c2²+b1²c2²+ b3²c2²+a3²Hb1²+b2²-2 b1 c1+c1²-2 b2 c2+c2²L-2 a1²b3 c3+

2 a1 b1 b3 c3+2 a1 b3 c1 c3-2 b1 b3 c1 c3-2 b2 b3 c2 c3+a1²c3²-

2 a1 b1 c3²+b1²c3²+b2²c3²+a2²Hb1²+b3²-2 b1 c1+c1²-2 b3 c3+c3²L- 2 a2H-b1 b2 c1+b2 c1²+a1Hb1-c1L Hb2-c2L+b1²c2+b3²c2-b1 c1 c2+

a3Hb2-c2L Hb3-c3L-b2 b3 c3-b3 c2 c3+b2 c3²L-2 a3Hb3 c1²-b2 b3 c2+ b3 c2²+a1Hb1-c1L Hb3-c3L+b1²c3+b2²c3-b2 c2 c3-b1 c1Hb3+c3LLLL, l ®HHb1²-a2 b2+b2²-a3 b3+b3²-b1 c1+a1H-b1+c1L+a2 c2-b2 c2+a3 c3-b3 c3L

Ha1²+a2²+a3²-2 a1 c1+c1²-2 a2 c2+c2²-2 a3 c3+c3²LL •

H2Ha1²b2²+a1²b3²-2 a1 b2²c1-2 a1 b3²c1+b2²c1²+b3²c1²-2 a1²b2 c2+ 2 a1 b1 b2 c2+2 a1 b2 c1 c2-2 b1 b2 c1 c2+a1²c2²-2 a1 b1 c2²+b1²c2²+ b3²c2²+a3²Hb1²+b2²-2 b1 c1+c1²-2 b2 c2+c2²L-2 a1²b3 c3+

a3Hb2-c2L Hb3-c3L-b2 b3 c3-b3 c2 c3+b2 c3²L-2 a3Hb3 c1²-b2 b3 c2+ b3 c2²+a1Hb1-c1L Hb3-c3L+b1²c3+b2²c3-b2 c2 c3-b1 c1Hb3+c3LLLL, m ®HHa1²+a2²+a3²-2 a1 b1+b1²-2 a2 b2+b2²-2 a3 b3+b3²L

Ha3 b3+a1Hb1-c1L-b1 c1+c1²+a2Hb2-c2L-b2 c2+c2²-a3 c3-b3 c3+c3²LL • H2Ha1²b2²+a1²b3²-2 a1 b2²c1-2 a1 b3²c1+b2²c1²+b3²c1²-2 a1²b2 c2+

2 a1 b1 b2 c2+2 a1 b2 c1 c2-2 b1 b2 c1 c2+a1²c2²-2 a1 b1 c2²+b1²c2²+ b3²c2²+a3²Hb1²+b2²-2 b1 c1+c1²-2 b2 c2+c2²L-2 a1²b3 c3+

a3Hb2-c2L Hb3-c3L-b2 b3 c3-b3 c2 c3+b2 c3²L-2 a3Hb3 c1²-b2 b3 c2+ b3 c2²+a1Hb1-c1L Hb3-c3L+b1²c3+b2²c3-b2 c2 c3-b1 c1Hb3+c3LLLL=, 9r® 1

€€€€2 ,HHHa1²+a2²+a3²-2 a1 b1+b1²-2 a2 b2+b2²-2 a3 b3+b3²L Ha1²+a2²+a3²-2 a1 c1+c1²-2 a2 c2+c2²-2 a3 c3+c3²L Hb1²+b2²+b3²-2 b1 c1+c1²-2 b2 c2+c2²-2 b3 c3+c3²LL •

Ha1²b2²+a1²b3²-2 a1 b2²c1-2 a1 b3²c1+b2²c1²+b3²c1²-2 a1²b2 c2+ 2 a1 b1 b2 c2+2 a1 b2 c1 c2-2 b1 b2 c1 c2+a1²c2²-2 a1 b1 c2²+b1²c2²+ b3²c2²+a3²Hb1²+b2²-2 b1 c1+c1²-2 b2 c2+c2²L-2 a1²b3 c3+

a3Hb2-c2L Hb3-c3L-b2 b3 c3-b3 c2 c3+b2 c3²L-2 a3Hb3 c1²-b2 b3 c2+ b3 c2²+a1Hb1-c1L Hb3-c3L+b1²c3+b2²c3-b2 c2 c3-b1 c1Hb3+c3LLLL, l ®HHb1²-a2 b2+b2²-a3 b3+b3²-b1 c1+a1H-b1+c1L+a2 c2-b2 c2+a3 c3-b3 c3L

Ha1²+a2²+a3²-2 a1 c1+c1²-2 a2 c2+c2²-2 a3 c3+c3²LL •

H2Ha1²b2²+a1²b3²-2 a1 b2²c1-2 a1 b3²c1+b2²c1²+b3²c1²-2 a1²b2 c2+ 2 a1 b1 b2 c2+2 a1 b2 c1 c2-2 b1 b2 c1 c2+a1²c2²-2 a1 b1 c2²+b1²c2²+ b3²c2²+a3²Hb1²+b2²-2 b1 c1+c1²-2 b2 c2+c2²L-2 a1²b3 c3+

a3Hb2-c2L Hb3-c3L-b2 b3 c3-b3 c2 c3+b2 c3²L-2 a3Hb3 c1²-b2 b3 c2+ b3 c2²+a1Hb1-c1L Hb3-c3L+b1²c3+b2²c3-b2 c2 c3-b1 c1Hb3+c3LLLL, m ®HHa1²+a2²+a3²-2 a1 b1+b1²-2 a2 b2+b2²-2 a3 b3+b3²L

L •