ans = -1.41421 1.00000

Lösungen

1

Ÿ a

ans = -1.41421 1.00000

Ÿ b

ans = 0.00019000

Ÿ c

Inf

Ÿ d

ans = 0 -3 2

Ÿ e

ans =

1.0e+01 *

3.40000 2.90000 6.00000 -4.76000 8.00000

Ÿ f

ans = 8.9000

2

Remove @ "Global`*" D

Ÿ a

u = H 3 a x + 2 y + 3 z Š 1 L ; v = H 2 x + 2 a y + 4 z Š 1 L ; w = H 3 x + 4 y + 1 a z Š 1 L ; Solve @8 u, v, w < , 8 x, y, z <D 99 x ® - -2 + 4 a - a

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

24 - 35 a + 3 a

, y ® - -9 + 14 a - 3 a

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

2 H 24 - 35 a + 3 a

L , z ® - -5 + 9 a - 3 a

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

24 - 35 a + 3 a

==

Solve @ 24 - 35 a + 3 a

Š 0, 8 a <D •• N

88 a ® 3. < , 8 a ® -3.71736 < , 8 a ® 0.717356 <<

Ÿ b

Solve @8 u, v, w < , 8 x, y, z <D • . a ® 1 99 x ® 1

€€€€ 8 , y ® 1

€€€€ 8 , z ® 1

€€€€ 8 ==

Ÿ c

v1 = 8 a 3, 2, 3 < ; v2 = 8 2, a 2, 4 < ; v3 = 8 3, 4, a 1 < ; v1 - v2 + 3 v3

8 7 + 3 a, 14 - 2 a, -1 + 3 a <

Det @8 v1, v2, v3 <D • . a ® 3 0

Keine Dim. da keine Lösung

Solve @8 u, v, w < , 8 x, y, z <D • . a ® 3

Power::infy : Infinite expression €€€€€1

0 encountered.

Mehr…

General::stop : Further output of Power::infy will be suppressed during this calculation.

Mehr…

88 x ® ComplexInfinity, y ® ComplexInfinity, z ® ComplexInfinity <<

Ÿ d

Solve @8 u, v < , 8 x, y, z <D • . a ® 2

Solve::svars : Equations may not give solutions for all "solve" variables.

Mehr…

99 x ® 1

€€€€€€€

10 - z

€€€€ 5 , y ® 1

€€€€ 5 - 9 z

€€€€€€€€€

10 ==

g @ t_ D := 9 1

€€€€€€€

10 - t

€€€€ 5 , 1

€€€€ 5 - 9 t

€€€€€€€€€

10 , t = ; P0 = 8 1, 1, 1 < ; v = g @ 1 D - g @ 0 D ; u = P0 - g @ 0 D ;

d = Norm @ Cross @ v, u DD • Norm @ v D 3 "########## €€€€€€€€

€€€€€€€€€€€€€€€€€€€€

2

N @ % D 1.56352

3

Ÿ a

v1 = 8 3, 2, 3 < ; v2 = 8 2, 2, 4 < ; v3 = 8 3, 4, 1 < ; w = 8 10, 12, 2 < ; Det @8 v1, v2, v3 <D

-16

Ÿ b

Solve @ w Š l1 v1 + l2 v2 + l3 v3, 8l1, l2, l3 <D 88 l1 ® 1, l2 ® -1, l3 ® 3 <<

4

Remove @ "Global`*" D

Ÿ a

a = 8 10, 10, 10 < ; e1 = 8 1, 0, 0 < ; e2 = 8 0, 1, 0 < ; e3 = 8 0, 0, 1 < ; LaengeHalbdiagonale = Norm @ a D

10 •!!!! 3

N @ % D 17.3205

Winkel Diagonale-Achse

WinkelDiagonaleAchse = ArcCos @ a.e1 • H Norm @ a D Norm @ e1 DLD ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

N @ % D 0.955317

a = N @ % D • Degree 54.7356

Test mit Richtungscosinusen o.k.:

solv = Solve @ 3 Cos @ x D ^ 2 == 1, 8 x <D •• Flatten

Solve::ifun : Inverse functions are being used by Solve, so some

solutions may not be found; use Reduce for complete solution information.

Mehr…

9 x ® -ArcCos A- 1

€€€€€€€€€€ •!!!! 3 E , x ® ArcCos A- 1

€€€€€€€€€€ •!!!! 3 E , x ® -ArcCos A 1

€€€€€€€€€€ •!!!! 3 E , x ® ArcCos A 1

€€€€€€€€€€ •!!!! 3 E=

solv4 = x • . solv @@ 4 DD ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

WinkelDiagonaleAchse == solv4 True

Skizze Würfel:

P1 = 8 1, 1, 1 < ; P2 = 8-1, 1, 1 < ; P3 = 8-1, -1, 1 < ; P4 = 8 1, -1, 1 < ; P5 = 8 1, 1, -1 < ; P6 = 8-1, 1, -1 < ; P7 = 8-1, -1, -1 < ; P8 = 8 1, -1, -1 < ; x1 = 8 1, 0, 0 < ; x2 = 8-1, 0, 0 < ; y1 = 8 0, 1, 0 < ;

y2 = 8 0, -1, 0 < ; z1 = 8 0, 0, 1 < ; z2 = 8 0, 0, -1 < ; linX = Line @8 x1, x2 <D ;

linY = Line @8 y1, y2 <D ; linZ = Line @8 z1, z2 <D ;

linO1 = Line @88 0, 0, 0 < , 8-0.75, 1, 1 <<D ; lin02 = Line @88 0, 0, 0 < , 8 1, 1, -0.75 <<D ; lin03 = Line @88 0, 0, 0 < , 8 1, -0.75, 1 <<D ; lin04 = Line @88 0, 0, 0 < , P1 <D ;

linP = 8 PointSize @ 0.03 D , Point @ P1 D , Point @8-0.75, 1, 1 <D ,

Point @8 1, 1, -0.75 <D , Point @8 1, -0.75, 1 <D , Point @8 0, 0, 0 <D ,

Line @8 P1, P2, P3, P4, P1, P5, P6, P2, P6, P7, P3, P7, P8, P4, P8, P5 <D , linX, linY, linZ, linO1, lin02, lin03, lin04 < ;

Show @ Graphics3D @ linP D ,

H*ViewPoint->8-1.424, 4.258, 2.660 <*L

ViewPoint -> 8 0.614, -1.560, 2.660 < , Boxed ® False D ;

Winkel zwischen zwei Diagonalen:

WinkelDiagonaleDiagonale = ArcCos @ P1.P2 • H Norm @ P1 D Norm @ P2 DLD ArcCos A 1

€€€€ 3 E

N @ % D 1.23096

b = N @ % D • Degree 70.5288

Winkel Kanten-Diagonalen:

ArcCos @ P1. H P1 - P2 L • H Norm @ P1 D Norm @ P1 - P2 DLD ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

N @ % D 0.955317

d = N @ % D • Degree 54.7356

Nochmals Winkel Kanten-Diagonalen:

g = H 180 - bL • 2 54.7356

WinkelKanteDiagonale = Pi - ArcCos @ P1. H P2 - P1 L • H Norm @ P1 D Norm @ P2 - P1 D LD p - ArcCos A- 1

€€€€€€€€€€ •!!!! 3 E

N @ % D 0.955317

d = N @ % D • Degree 54.7356

Ÿ b

Winkel Kante Achse

Diagonalwinkel mit Achse : ArcCos A 1

€€€€€€€€€ •!!!! 3 E , Winkel Kanten - Diagonalen : p - ArcCos A - 1

€€€€€€€€€ •!!!! 3 E oder ArcCos A 1

€€€€€€€€€ •!!!! 3 E

9p - ArcCos A- 1

€€€€€€€€€€ •!!!!

3

E , ArcCos A 1

€€€€€€€€€€ •!!!!

3

E= •• N

8 0.955317, 0.955317 <

WinkelKanteAchse = Pi - WinkelKanteDiagonale - WinkelDiagonaleAchse ArcCos A- 1

€€€€€€€€€€ •!!!! 3 E - ArcCos A 1

€€€€€€€€€€ •!!!! 3 E N @ % D

1.23096

WinkelKanteAchse = Pi - 2 WinkelDiagonaleAchse •• Simplify p - 2 ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

N @ % D 1.23096

N @ % D • Degree 70.5288

H Sin @ WinkelKanteAchse DL •• N 0.942809

LaengeKanteAchse =

LaengeHalbdiagonale • Sin @ WinkelKanteAchse D * Sin @ WinkelDiagonaleAchse D 10 •!!!! 2 Csc A 2 ArcCos A 1

€€€€€€€€€€ •!!!! 3 EE

N @ % D 15.

VolumenEckstueck1 = LaengeKanteAchse^ 3 • 6

€€€€€€€€€€€€€ 1000 3

•!!!! 2 Csc A 2 ArcCos A 1

€€€€€€€€€€ •!!!! 3 EE

N @ % D 562.5

VKoerper = 2 VolumenEckstueck1

€€€€€€€€€€€€€ 2000 3

•!!!! 2 Csc A 2 ArcCos A 1

€€€€€€€€€€ •!!!! 3 EE

N @ % D 1125.

Ÿ Andere Variante

v1 @ t_ D := a - t e1; v2 @ t_ D := a - t e2; Solve @ v1 @ t D .v2 @ t D Š 0, 8 t <D 88 t ® 15 <<

v1 @ t D .v2 @ t D 100 + 20 H 10 - t L v1 @ 15 D

8-5, 10, 10 <

v2 @ 15 D 8 10, -5, 10 <

Norm @ v1 @ 15 DD 15

8 15 e1, 15 e2, 15 e3 <

88 15, 0, 0 < , 8 0, 15, 0 < , 8 0, 0, 15 <<

H 15 e1 - a L . H 15 e2 - a L 0

H 15 e1 - a L . H 15 e3 - a L 0

H 15 e2 - a L . H 15 e3 - a L 0

VolumenEckstueck2 = 15 ^ 3 • 6

€€€€€€€€€€€€€ 1125 2 N @ % D 562.5

VolumenKoerper = 2 VolumenEckstueck2 1125

Kontrollen:

WinkelKanteAchse = ArcCos @ v1 @ 15 D .e1 • Norm @ v1 @ 15 DDD ArcCos A- 1

€€€€ 3 E

N @ % D 1.91063

N @ % D • Degree 109.471

180 - % 70.5288

Sin @ % Degree D

0.942809

Sin @ % D 0.809212

e @l_ D := l e1; e @lD . H a - e @lDL Š 0 H 10 - l L l Š 0

Solve @H 10 - lL l Š 0, 8l<D 88l ® 0 < , 8l ® 10 <<

Ÿ c

kantenlaenge = 2 LaengeHalbdiagonale • Sqrt @ 3 D 20

Ÿ d

VolumenKoerper = 2 VolumenEckstueck2 1125

5

Remove @ "Global`*" D

p1 = 8 4, 1 < ; p2 = 8 2, 3 < ; j = 68.44 Degree;

Ÿ a

m = 88 Cos @ j D , -Sin @ j D< , 8 Sin @ j D , Cos @ j D<< ; m •• MatrixForm J 0.367475 -0.930033

0.930033 0.367475 N p3 = m.p2

8-2.05515, 2.96249 <

Ÿ b

Det @8 p2 - p1, p2 + H p3 - p1 L<D

-1.81469

6

Remove @ "Global`*" D

r1 = 8 1, 0, 1 < ; r2 = 8 1, 2, -1 < ; a1 = 8 4, -1, 1 < ; a2 = 8-1, -1, 6 < ; p0 = 8 10, 1, -2 < ;

Ÿ a

Vol = Det @8 a1, a2, r2 - r1 <D -40

Š> Schneiden sich nicht

Solve @ a1 Š a a2, 8a<D 8<

Š> windschief

Ÿ b

d = Vol • H Norm @ Cross @ a1, a2 DDL

- 8

€€€€€€€€€€€€€€

3 •!!!! 3 N @ % D -1.5396

Ÿ c

d = Det @8 a1, a2, p0 - r1 <D • H Norm @ Cross @ a1, a2 DDL

- 11

€€€€€€€€€€€€€€

3 •!!!! 3 N @ % D -2.11695

7

Remove @ "Global`*" D

p1 = 8 4, 1 < ; p2 = 8 2, 3 < ; p3 = 8 -6, -2 < ;

Ÿ a

k @ x_, y_, r_ D := H8 x, y < - p1 L . H8 x, y < - p1 L - r ^ 2;

k @ vec_, r_ D := H vec - p1 L . H vec - p1 L - r ^ 2 k @ p2, r D

8 - r

Solve @ k @ p2, r D Š 0, 8 r <D 99 r ® -2 •!!!! 2 = , 9 r ® 2 •!!!! 2 ==

N @ % D

88 r ® -2.82843 < , 8 r ® 2.82843 <<

k @ vec_ D := k A vec, 2 •!!!! 2 E p4 = H p3 + p1 L • 2

9 -1, - 1

€€€€ 2 =

r1 = Norm @ p1 - p4 D

•!!!!!!!!!! 109

€€€€€€€€€€€€€€€

2 N @ % D 5.22015

k1 @ vec_ D := H vec - p4 L . H vec - p4 L - r1 ^ 2 8 k @8 x, y <D == 0, k1 @8 x, y <D Š 0 <

9-8 + H-4 + x L

+ H-1 + y L

Š 0, - 109

€€€€€€€€€€

4 + H 1 + x L

+ J 1

€€€€ 2 + y N

Š 0 =

Solve @8 k @8 x, y <D == 0, k1 @8 x, y <D Š 0 < , 8 x, y <D

99 x ® 2

€€€€€€€€€€

109 I 178 - 3 •!!!!!!!!!! 202 M , y ® 5

€€€€€€€€€€

109 I 17 + 4 •!!!!!!!!!! 202 M= ,

9 x ® 2

€€€€€€€€€€

109 I 178 + 3 •!!!!!!!!!! 202 M , y ® 5

€€€€€€€€€€

109 I 17 - 4 •!!!!!!!!!! 202 M==

solv = N @ % D

88 x ® 2.48371, y ® 3.38765 < , 8 x ® 4.0484, y ® -1.82801 <<

solv @@ 1 DD

8 x ® 2.48371, y ® 3.38765 <

p5 = 8 x, y < • . solv @@ 1 DD ; T1 = p5

8 2.48371, 3.38765 <

p6 = 8 x, y < • . solv @@ 2 DD ; T2 = p6 8 4.0484, -1.82801 <

<< Graphics`ImplicitPlot`

pl = ImplicitPlot @8 k @8 x, y <D Š 0, k1 @8 x, y <D Š 0 < , 8 x, -7, 7 <D ; Prepend @ Map @ Point, 8 p1, p2, p3, p4, p5, p6 <D , PointSize @ 0.04 DD 9 PointSize @ 0.04 D , Point @8 4, 1 <D , Point @8 2, 3 <D , Point @8-6, -2 <D ,

Point A9-1, - 1

€€€€ 2 =E , Point @8 2.48371, 3.38765 <D , Point @8 4.0484, -1.82801 <D=

lin = Show @ Graphics @ Line @8 p5, p3, p1, p5, p1, p6, p3 <DDD ;

gra = Show @ Graphics @ Prepend @ Map @ Point, 8 p1, p2, p3, p4, p5, p6 <D , PointSize @ 0.04 DDD , pl, AspectRatio ® Automatic D ;

Show @ gra, lin D ;

Ÿ b

len = Norm @ p3 - p6 D 10.0499

Norm @ p3 - p5 D 10.0499

Ÿ c

len ^ 2 Š H s + 1 L * s 101. Š s H 1 + s L

solv = Solve @ len ^ 2 Š H s + 1 L * s, 8 s <D

88 s ® -10.5623 < , 8 s ® 9.56231 <<

s = s • . solv @@ 2 DD 9.56231

s * H s + 1 L Š len ^ 2

True