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J Cos @jD -Sin @jD

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1

Remove @ "Global`*" D

Dreh @j_ D := 88 Cos @jD , -Sin @jD< , 8 Sin @jD , Cos @jD<< ; Dreh @jD •• MatrixForm

J Cos @jD -Sin @jD

Sin @jD Cos @jD N

P1 = 8 2, 1 < ; P2 = 8 3, 2 < ; P3 = 8 1, 3 < ; j = 71 Degree;

Q1 = Dreh @jD .P1 •• N

8 -0.294382, 2.21661 <

Q2 = Dreh @jD .P2 •• N

8-0.914333, 3.48769 <

Q3 = Dreh @jD .P3 •• N

8-2.51099, 1.92222 <

(2)

o = 8 0, 0 < ; Show @ Graphics @

8

Line @88-1, 0 < , 8 1, 0 <<D , Line @88 0, -1 < , 8 0, 1 <<D , Line @8 P1, P2, P3, P1 <D , Line @8 Q1, Q2, Q3, Q1 <D , Line @8 o, P1 <D , Line @8 o, P2 <D , Line @8 o, P3 <D , Line @8 o, Q1 <D , Line @8 o, Q2 <D , Line @8 o, Q3 <D , PointSize @ 0.03 D , Point @ P1 D ,

Point @ P2 D , Point @ P3 D , Point @ Q1 D , Point @ Q2 D , Point @ Q3 D

<

D , AspectRatio ® Automatic D ;

2

a@ x_ D := ArcTan @ x @@ 2 DD • x @@ 1 DDD x1 = 8 4, 1 < ; a@ x1 D

ArcTan A 1

€€€€ 4 E

% •• N 0.244979

S @ 0 D = 88 1, 0 < , 8 0, -1 << ; S @ 0 D •• MatrixForm

J 1 0

0 -1 N

S @a_ D := Dreh @aD .S @ 0 D .Dreh @-aD ; S @aD •• Simplify •• MatrixForm J Cos @ 2 aD Sin @ 2 aD

Sin @ 2 a D -Cos @ 2 a D N S1 = S @a@ x1 DD .P1 9 38

€€€€€€€

17 , 1

€€€€€€€

17 =

(3)

S1 •• N

8 2.23529, 0.0588235 <

S2 = S @a@ x1 DD .P2 9 61

€€€€€€€

17 , - 6

€€€€€€€

17 = S2 •• N

8 3.58824, -0.352941 <

S3 = S @a@ x1 DD .P3 9 39

€€€€€€€

17 , - 37

€€€€€€€

17 = S2 •• N

8 3.58824, -0.352941 <

Show @ Graphics @ 8

Line @88 -1, 0 < , 8 1, 0 <<D , Line @88 0, -1 < , 8 0, 1 <<D , Line @8 P1, P2, P3, P1 <D , Line @8 S1, S2, S3, S1 <D , Line @8 o, x1 <D ,

Line @8 P1, S1 <D , Line @8 P2, S2 <D , Line @8 P3, S3 <D , PointSize @ 0.03 D , Point @ P1 D ,

Point @ P2 D , Point @ P3 D , Point @ S1 D , Point @ S2 D , Point @ S3 D

D < , AspectRatio ® Automatic D ;

(4)

3

Eigensystem @ S @a@ x1 DDD 98 -1, 1 < , 99 - 1

€€€€ 4 , 1 = , 8 4, 1 <==

8 4, 1 < ist x1 gestreckt H Richtung der Spiegelungsgerade L . 9 - 1

€€€€€

4 , 1 = ist darauf senkrecht. Der Eigenwert - 1 bewirkt die Spiegelung, der Eigenwert 1 das Fixhalten der Abbildung senkrecht zur Speigelungsachse.

4

Remove @ "Global`*" D vec0 = 8 0, 0, 0 < ;

a = 8 3, 1, 2 < ; b = 8-1, 2, -2 < ; u = 8 2, 3, -1 < ; M = Transpose @8 a, b, u <D ; Det @ M D -7

a, b, u sind linear unabhängig, da die Determinante nicht 0 ist.

Proj.Transpose[{a,b,u}]=Transpose[{a,b,0}] Š> Proj = Transpose[{a,b,0}].Inverse[M]

M •• MatrixForm i

k jjjjj jj

3 -1 2

1 2 3

2 -2 -1 y { zzzzz zz

Transpose @8 a, b, vec0 <D •• MatrixForm i

k jjjjj jj

3 -1 0

1 2 0

2 -2 0 y { zzzzz zz

vec0 = 8 0, 0, 0 < ;

Proj = Transpose @8 a, b, vec0 <D .Inverse @ M D ; Proj •• MatrixForm i

k jjjjj jjjjj

- €€€€

57

€€€€

87

2 - €€€€€€

187

€€€€€€

197

3

€€€€

67

- €€€€

47

0 y

{ zzzzz zzzzz

% •• N •• MatrixForm i

k jjjjj jj

-0.714286 1.14286 2.

-2.57143 2.71429 3.

0.857143 -0.571429 0.

y { zzzzz zz

T1 = 8 0, 2, 3 < ; T2 = 8 1, 1, 0 < ; T3 = 8 2, 0, 2 < ; N1 = Proj.T1

9 58

€€€€€€€

7 , 101

€€€€€€€€€€

7 , - 8

€€€€ 7 =

(5)

% •• N

8 8.28571, 14.4286, -1.14286 <

N2 = Proj.T2 9 3

€€€€ 7 , 1

€€€€ 7 , 2

€€€€ 7 =

% •• N

8 0.428571, 0.142857, 0.285714 <

N3 = Proj.T3 9 18

€€€€€€€

7 , 6

€€€€ 7 , 12

€€€€€€€

7 =

% •• N

8 2.57143, 0.857143, 1.71429 <

o = 8 0, 0, 0 < ; Show @ Graphics3D @

8

Line @88 -1, 0, 0 < , 8 1, 0, 0 <<D ,

Line @88 0, -1, 0 < , 8 0, 1, 0 <<D , Line @88 0, 0, -1 < , 8 0, 0, 1 <<D , Line @8 N1, N2, N3, N1 <D , Line @8 T1, T2, T3, T1 <D ,

Line @8 o, u <D ,

Line @8 T1, N1 <D , Line @8 T2, N2 <D , Line @8 T3, N3 <D , PointSize @ 0.02 D , Point @ T1 D , Point @ T2 D ,

Point @ T3 D , PointSize @ 0.035 D , Point @ N1 D , Point @ N2 D , Point @ N3 D

D < , AspectRatio ® Automatic, PlotRange ® 88-1, 9 < , 8-1, 15 < , 8-2, 4 << D ;

(6)

5

M1 = H T1 + N1 L • 2 9 29

€€€€€€€

7 , 115

€€€€€€€€€€

14 , 13

€€€€€€€

14 = N @ % D

8 4.14286, 8.21429, 0.928571 <

M2 = H T2 + N2 L • 2 9 5

€€€€ 7 , 4

€€€€ 7 , 1

€€€€ 7 =

N @ % D

8 0.714286, 0.571429, 0.142857 <

M3 = H T3 + N3 L • 2 9 16

€€€€€€€

7 , 3

€€€€ 7 , 13

€€€€€€€

7 = N @ % D

8 2.28571, 0.428571, 1.85714 <

Matr = Transpose @8 M1, M2, M3 <D .Inverse @ Transpose @8 T1, T2, T3 <DD ; Matr •• MatrixForm i

k jjjjj jjjjj

€€€€

17

€€€€

47

1 - €€€€

97

€€€€€€

137

€€€€

32

€€€€

37

- €€€€

27

€€€€

12

y

{ zzzzz zzzzz

N @ % D •• MatrixForm i

k jjjjj jj

0.142857 0.571429 1.

-1.28571 1.85714 1.5

0.428571 -0.285714 0.5 y { zzzzz zz

Eigensystem @ Matr D •• N

88 1., 1., 0.5 < , 88 1.16667, 0., 1. < , 8 0.666667, 1., 0. < , 8 -2., -3., 1. <<<

a={3,1,2}; b={-1,2,-2}; u={2,3,-1};

Auf den Ersten Blick fällt auf, dass unter den Eigenvektoren -u und daher u (gestreckt) vorkommt. Der dazugehörige Eigenwert ist 0.5, während die andern beiden 1 sind (keine Streckung in diese Richtungen).

Matr.T1 9 29

€€€€€€€

7 , 115

€€€€€€€€€€

14 , 13

€€€€€€€

14 =

Richtig

(7)

Matr.T2 9 5

€€€€ 7 , 4

€€€€ 7 , 1

€€€€ 7 =

Richtig

Matr.T3 9 16

€€€€€€€

7 , 3

€€€€ 7 , 13

€€€€€€€

7 = Richtig

6

Remove @ "Global`*" D a = 8 3, 1, 2 < ; b1 = a • Norm @ a D

9 3

€€€€€€€€€€€€€ •!!!!!!! 14 , 1

€€€€€€€€€€€€€ •!!!!!!! 14 , $%%%%%%% 2

€€€€ 7 =

8 1, y1, 0 < .a Š 0 3 + y1 Š 0

Solve @8 1, y1, 0 < .a Š 0, 8 y1 <D 88 y1 ® -3 <<

b2work = 8 1, -3, 0 <

8 1, -3, 0 <

b2 = b2work • Norm @ b2work D

9 1

€€€€€€€€€€€€€ •!!!!!!! 10 , - 3

€€€€€€€€€€€€€ •!!!!!!! 10 , 0 =

b3 = Cross @ b1, b2 D

9 3

€€€€€€€€€€€€€ •!!!!!!! 35 , 1

€€€€€€€€€€€€€ •!!!!!!! 35

, -$%%%%%%% 5

€€€€ 7 =

U = Transpose @8 b1, b2, b3 <D ; U •• MatrixForm i

k jjjjj jjjjj jjjj

€€€€€€€€€€

•!!!!!!!!143

€€€€€€€€€€

•!!!!!!!!101

€€€€€€€€€€

•!!!!!!!!353

€€€€€€€€€€

•!!!!!!!!141

- €€€€€€€€€€

•!!!!!!!!310

€€€€€€€€€€

•!!!!!!!!351

"###### €€€€

27

0 - "###### €€€€

57

y

{

zzzzz

zzzzz

zzzz

(8)

Dreh0 @j_ D := 88 1, 0, 0 < , 8 0, Cos @jD , -Sin @jD< , 8 0, Sin @jD , Cos @jD<< ; Dreh0 @jD •• MatrixForm

i k jjjjj jj

1 0 0

0 Cos @jD -Sin @jD 0 Sin @jD Cos @jD

y { zzzzz zz

DrehAchse @ j_ D := U.Dreh0 @ j D .Inverse @ U D

Dreh = DrehAchse @ 56 Degree D •• N; Dreh •• MatrixForm i

k jjjjj jj

0.842569 -0.348681 0.410487 0.537598 0.590679 -0.601736

-0.0326523 0.727681 0.685138

y { zzzzz zz

T1 = 8 0, 2, 3 < ; T2 = 8 1, 1, 0 < ; T3 = 8 2, 0, 2 < ; R1 = Dreh.T1

8 0.5341, -0.623851, 3.51078 <

R2 = Dreh.T2

8 0.493888, 1.12828, 0.695029 <

R3 = Dreh.T3

8 2.50611, -0.128277, 1.30497 <

(9)

o = 8 0, 0, 0 < ; Show @ Graphics3D @

8

Line @88-1, 0, 0 < , 8 1, 0, 0 <<D ,

Line @88 0, -1, 0 < , 8 0, 1, 0 <<D , Line @88 0, 0, -1 < , 8 0, 0, 1 <<D , Line @8-4 b1, 4 b1 <D , Line @8-2 b2, 2 b2 <D , Line @8-2 b3, 2 b3 <D , Line @8 R1, R2, R3, R1 <D , Line @8 T1, T2, T3, T1 <D ,

PointSize @ 0.02 D , Point @ T1 D , Point @ T2 D ,

Point @ T3 D , PointSize @ 0.035 D , Point @ R1 D , Point @ R2 D , Point @ R3 D

<

D , AspectRatio ® Automatic D ;

7

Ÿ a Berechnung der Punkte

Remove @ "Global`*" D a = 8 3, 1, 2 < ; b = 8-1, 3, 0 < ; c = Cross @ a, b D 8 -6, -2, 10 <

B = Transpose @8 a, b, c <D ; B •• MatrixForm i

k jjjjj jj

3 -1 -6 1 3 -2 2 0 10

y

{ zzzzz

zz

(10)

Dl = 88 1, 0, 0 < , 8 0, 1, 0 < , 8 0, 0, -1 << ; Dl •• MatrixForm i

k jjjjj jj

1 0 0 0 1 0 0 0 -1

y { zzzzz zz

A = B.Dl.Inverse @ B D ; A •• MatrixForm i

k jjjjj jjjjj

€€€€€€

1735

- €€€€€€

356

€€€€

67

- €€€€€€

356

€€€€€€

3335

€€€€

27

€€€€

67

€€€€

27

- €€€€

37

y

{ zzzzz zzzzz

T1 = 8 0, 2, 3 < ; T2 = 8 1, 1, 0 < ; T3 = 8 2, 0, 2 < ; S1 = A.T1

9 78

€€€€€€€

35 , 96

€€€€€€€

35 , - 5

€€€€ 7 =

S2 = A.T2 9 11

€€€€€€€

35 , 27

€€€€€€€

35 , 8

€€€€ 7 =

S3 = A.T3 9 94

€€€€€€€

35 , 8

€€€€€€€

35 , 6

€€€€ 7 =

M1 = H T1 + S1 L • 2 9 €€€€€€€ 39

35 , €€€€€€€ 83 35 , €€€€ 8

7 = M2 = H T2 + S2 L • 2 9 23

€€€€€€€

35 , 31

€€€€€€€

35 , 4

€€€€ 7 =

M3 = H T3 + S3 L • 2 9 82

€€€€€€€

35 , 4

€€€€€€€

35 , 10

€€€€€€€

7 =

(11)

o = 8 0, 0, 0 < ; Show @ Graphics3D @

8

Line @88-1, 0, 0 < , 8 1, 0, 0 <<D ,

Line @88 0, -1, 0 < , 8 0, 1, 0 <<D , Line @88 0, 0, -1 < , 8 0, 0, 1 <<D ,

Line @8 S1, S2, S3, S1 <D , Line @8 T1, T2, T3, T1 <D , Line @8 M1, M2, M3, M1 <D , Line @8 S1, T1 <D , Line @8 S2, T2 <D , Line @8 S3, T3 <D ,

PointSize @ 0.03 D , Point @ T1 D , Point @ T2 D , Point @ T3 D ,

PointSize @ 0.05 D , Point @ S1 D , Point @ S2 D , Point @ S3 D , PointSize @ 0.07 D , Point @ M1 D , Point @ M2 D , Point @ M3 D , PointSize @ 0.09 D , Point @ o D

<

D , AspectRatio ® Automatic, ViewPoint -> 8 0.478, 4.341, 0.259 < D ;

Ÿ b Kontrolle: Liegen die Punkte M

k

in der Ebene F?

Solve @l a + m b + n c == M1, 8l, m, n<D

99l ® 4

€€€€ 7 , m ® 3

€€€€ 5 , n ® 0 ==

Solve @l a + m b + n c == M2, 8l, m, n<D

99 l ® 2

€€€€ 7 , m ® 1

€€€€ 5 , n ® 0 ==

Solve @l a + m b + n c == M3, 8l, m, n<D

99l ® 5

€€€€ 7 , m ® - 1

€€€€ 5 , n ® 0 ==

In Ordnung, denn der Koeffizient n für die Richtung c ist immer 0.

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