Coordinate Transformations Lecture Notes

Introduction

• This session focuses on coordinate transformations, specifically rotations.
• Previous session covered translations (origin shift in x and y directions with parallel axes).
• Focus: Transforming coordinates from original to rotated system (origin unchanged, axes rotated).

Original Coordinate System

• Start by drawing original x and y axes.
• Example vector has x-component = 1 and y-component = 2.

Rotated Coordinate System

• New axes: x' and y', rotated by an angle alpha from original x-axis.
• Objective: Find x' and y' components for the vector in the rotated system.

Key Concepts

• Rotation of Axes: Rotating coordinate axes by alpha is equivalent to rotating vector by -alpha.
• In original system, x-component < y-component (vector mainly in y-direction).
• In rotated system, x-component > y-component (vector mainly in x'-direction).

Rotation Matrix

• Derive using concept: rotating vector by -alpha when axes rotate by alpha.
• Rotation Matrix: Using cosine and sine properties:
  • cos(-alpha) = cos(alpha)
  • sin(-alpha) = -sin(alpha)
• Matrix:
  • [cos(alpha), sin(alpha); -sin(alpha), cos(alpha)]

Example Calculation

1. Angle Alpha: Given as 30 degrees (or π/6).
2. Rotation Matrix with alpha = 30 degrees:
   • [sqrt(3)/2, 1/2; -1/2, sqrt(3)/2]
3. Multiply with original vector (1, 2) to find x' and y'.
   • Resulting vector in rotated reference: approx (1.86, 1.22).

Application in New Exercise

• Initial x and y components both = 1.
• Rotate coordinate system by angle vector makes with original system.
• Objective: Find x' and y' in new system.

Finding Angle Alpha

• Alpha = angle vector makes with original x-axis; also rotation angle.
• Use tangent formula: tan(alpha) = y/x
  • Here, tan(alpha) = 1, leading to alpha = 45 degrees.

Rotation Matrix for New Exercise

• Angle to use: -45 degrees.
• Matrix:
  • [cos(45), sin(45); -sin(45), cos(45)]
• Cos(45) = Sin(45) = sqrt(2)/2.
• Using rotation matrix on vector (1, 1):
  • x' = sqrt(2)
  • y' = 0

Conclusion

• Understanding of rotating coordinate axes using rotation matrix.
• Use rotation matrix for angle opposite in sign to axis rotation angle.
• Questions and suggestions welcome.
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