This document describes various mathematical and geometric concepts including: 1) Defining a 3D coordinate system with points (0,0,0), (1,0,0), (1,1,0), (0,1,0) and describing a rotation matrix R that rotates this coordinate system. 2) Discussing the cross product of two vectors x and y and defining it as t = qc - rc. 3) Providing equations for representing a rotation matrix R in terms of unit vectors x, y, z and the rotated unit vectors xr, yr, zr.

1. AR
@
A 1
4

2. R
A
2

3. A
3

4. A
4

5. 5

6. AR 4
6

7. :
7/ / . . / .- .-

8. 8
X
Z
Y
A
4
[R | t]
M

9. 9

10. 10

11. □ 2
11

12. 12
x
y

13. D 102 1 2
= 2
10 , ( 4
( ,- , ( ,- , :
(
2 A
= :
2 =
13) R
(0,0,0)
(1,0,0)
(1,1,0)
(0,1,0)

14. 1 ]K
A
[ = 2R
4 2 - - ,
- - , () H
14c
qz=0

15. 15

16. A
16

17. A
17

18. A
4d
4 e
- A
R
18

19. 3 3
3
B
3
19

20. 7 :
20/ .

21. □
21
12 12

22. 2
4 42
4 1
221
A A

23. 1
□
□
□ 2 2
23

24. 2
2 4-
(
1 2
R A
)
2
2
242

25. u V m i
rkd V md e
□ ap s M T
□ g rkd f S K
D V mP h Oto
ln cR T
250 0 2 0A 0 , 2 2B 0 2 B
0 0 8 -B ,-

26. 26
r1
q1

27. 3
2 1
27
X
Z
Y
X
Z
Y
t = qc - rc

28. 2
31
28
X
Y
Z
x
y
z
||x|| = ||y|| = ||z|| = 1

29. 32
29
X
Y
Z
x
y
z
1
X
Y
Z
[ x | y | z ]

30. 2 1
3
30
X
Y
Z
x
y
z
X
Y
Z
[ x | y | z ]
X
Y
Z
xr
yr
zr
3
[ xr | yr | zr ]
R = [ xr | yr | zr ] [ x | y | z ]T

31. A
31

32. A
32

33. _ f hak T
Of ci g
hak T I FC
V S
jmO T lC
33
2 . . :2 /. :. - 2/ 2 : 2/ 2 :
akd e

34. .
t e
c
34

35. □
□
35

36. 36

37. r e [
v r e
□ f D tpEi P
r eI z noul
d IFG
□ F a
□ ]
□ V F
□ s T g
c k . P, F
37, 1 . . . 2 1 1 1 1 . 0 1. 1 1 2 . .

38. :: oc
c d H oh
en m
l m pa g b
□ p b
i
383 9 780 6 3 3 17 .582 /86 17 :

39. .0 7 bgd
I e c
a
I/ / i
39hIH 6 : 309 67 60 :018 5 2 9 : ) ) (( ,(

40. 1
H 1
1
1 2 1
40

41. c
C
a _ H
V O
V
d
41
/:: . : / 2
: : - : - : / - : / /:

42. 2
□
□ 2
□ 2
,
42

43. 0
□
□
□
43

44. □
□
44

45. DoG
45
13 3 3 2 / 20/0 03 . 2 3

46. 2 D
: D G
46
3 / 1 212 . 2 0

47. 1
474
2

48. F
r
T o
A S
48

49. □
49

50. D T T
G A F
4 I T
50R S

51. 6 F
= = 6
= 6 S
= 4 2
= =
I6 T 18
51

52. □
□
01
8
2 2
%
52

53. 1 I
S 1 T 1
A1
. 4 R
4
531F

54. A 4 .
4
2 R
54

55. ④ ]
2 6
A
3 P
□ 3 A
1
- - A
□ 4 D
c
55[ R

56. P
□
4
56

57. .
1
. 2
57

58. S A
□ 3
C/ ] /
R
□ ] 3
/ [
/
□ [ /
□ N 58

59. M
M
59
Tukey

60. 60

61. c s
R
MR
□
c
n
n
M a
61