1. zB B yB zA B0
VA0B0 A xB yA A0 xA
Figure 2.1: Two Coordinate Systems.

2. B A zB 1 yB VB01 zA VA01 B0 VA0B0 xB yA A0 xA
Figure 2.2: Depiction of Point Transformation Problem.

3. zB 1 B b3 yB b1 b2 xB
Figure 2.3: Point 1 Projected onto the B Coordinate System

4. C B A zC 1 VC01 yC C0 xC VA01 zB VB0C0 zA yB B0 VA0B0 xB A0 yA xA
Figure 2.4: Compound Transformation

5. zA zB yB  yA xA
Figure 2.5: Rotation About the X Axis

6. Coordinate system B is initially aligned with coordinate system A.
It is then translated to the point [5, 4, 1]T.
It is then rotated 30 about its X axis.
It is then rotated 60 about an axis that passes through the point [2, 0, 2]T, measured in the current coordinate system, which is parallel to the current Y axis.
Find

7. zB zA yB B A xA yA xB
Figure 2.6: Initial and Final Coordinate Systems

8. A C C initially aligned with A ; translated to [5,4,1]T zA zC xA yA xCyC

9. C D D initially aligned with C ; rotated 30° about the X axis zC zD yDxC yC

10. B D B initially aligned with D ; rotated 60° about an axis that passes
through point M, [2, 0, 2]T and is parallel to the Y axis zB yB B zD
60° D M yD xB xD

11. E initially aligned with D ; translate to point M, i.e. [2, 0, 2]TzE zB yB E B
60° zF zD yE F xE D M yD xB xF xD

12. E B F D F initially aligned with E ; rotate 60° about the Y axis zE zByB E B
60° zF zD yE F xE D M yD xB xF xD

13. E B F D B initially aligned with F ; translate to [-2, 0, -2]T zE zByB E B
60° zF zD yE F xE D M yD xB xF xD

14. m  zA zB yB yA xA xB
Figure 2.8: Rotation of Angle  about axis m

15. C and A will rotate together by an angle  about m
A moves to B ; C moves to D m zA
the relationship between C and A is the same as the relationship between D and B zC yC yA xA xC

16. Coordinate system B is initially aligned with coordinate system A.
It is then rotated an angle  about the axis m (unit vector).
Find m zA  yA xA

17. Coordinate system C is created such that its Z axis is along the direction of m.zC yA yC xA xC

18. Coordinate systems A and C rotate together about m by the angle .
A rotates to become B and C rotates to become D.
The relationship between B and D must be the same as the relationship between A and C. m zA zC  yA yC xA xC

19. Lastly, the rotation matrix that relates the B and A coordinate systems may be written aszA zC  yA yC xA xC

21. 1st row is a unit vector
ax2 + bx2 + mx2 = 1
ax2 + bx2 = 1 - mx2
1st and 2nd rows are orthogonal
axay + bxby+ mxmy = 0
3rd column is cross product of 1st two columns
mz = axby - bxay

23. given:
find: m and 

24. r11 + r22 + r33 = (1-cos) (mx2+my2+mz2) + 3 cos
r11 + r22 + r33 =1 - cos + 3 cos
r21 - r12 = 2 mz sin
r13 - r31 = 2 my sin
r32 - r23 = 2 mx sin