Can this represent rotation? Why?

  • Context: Graduate 
  • Thread starter Thread starter yicong2011
  • Start date Start date
  • Tags Tags
    Rotation
Click For Summary
SUMMARY

The discussion centers on the mathematical representation of rotation using complex numbers, specifically through the equations r² = (x - ia)² + (y - ib)² + (z - ic)² and r² = x² + y² + (z - ia)². The use of the imaginary unit i (where i² = -1) is crucial in defining rotation in three-dimensional space. The equations illustrate how displacement and rotation can be represented differently, emphasizing the need for a deeper understanding of these mathematical concepts to grasp their implications in Euclidean space.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with Euclidean geometry
  • Knowledge of mathematical notation and equations
  • Basic concepts of rotation in three-dimensional space
NEXT STEPS
  • Explore the implications of complex numbers in physics and engineering
  • Study the mathematical foundations of rotation matrices
  • Learn about quaternion representation of rotations
  • Investigate the relationship between complex numbers and transformations in 3D space
USEFUL FOR

Mathematicians, physicists, and engineers interested in the mathematical foundations of rotation and displacement in three-dimensional space.

yicong2011
Messages
75
Reaction score
0
If we say

r2= (x-a)2 + (y-b)2 + (z-c)2

can generally represent displacement,

why can

r2= (x- ia)2 + (y- ib)2 + (z- ic)2

generally represent rotation?

(i2 = -1)


And why

r2= x2 + y2 + (z - ia)2

represents rotation with respect to z axis
 
Physics news on Phys.org

Similar threads

Prove a rational fraction is equal to another
  • · Replies 5 ·
Replies
5
Views
2K
Engineering Stiffness of a Concrete Core in a Multi-Storey Building
  • · Replies 2 ·
Replies
2
Views
3K
MHB How is z=2xy a Hyperbolic Paraboloid in the rotated 45° in the xy-plane?
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
LaTeX Typesetting standards for subscripts
  • · Replies 5 ·
Replies
5
Views
4K
How Does Coriolis Force Influence Particle Motion in Rotating Systems?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Curl of velocity vector in rotational motion
  • · Replies 9 ·
Replies
9
Views
951
Detector moving toward charged ring - rate of change of V
  • · Replies 5 ·
Replies
5
Views
1K
A line integral about a closed "unit triangle" around the origin
  • · Replies 12 ·
Replies
12
Views
5K
Graduate Displacement-Rotation Algorithm
  • · Replies 2 ·
Replies
2
Views
2K