Conversion of Cartesian Vector into Cylindrical Vector

Introduction

• Overview of previous videos:
  • Explained cylindrical coordinate system.
  • Conversion of cylindrical coordinates to Cartesian coordinates.
  • Conversion of Cartesian coordinates to cylindrical coordinates.
• Objective of this video:
  • Convert Cartesian vector into cylindrical vector.
  • Solve a problem based on this conversion.

Understanding Cartesian and Cylindrical Vectors

• Cartesian Vector (P):
  • P = ax * i_x + a_y * i_y + a_z * i_z
• Cylindrical Parameters:
  • R (radius), φ (phi, angle), Z (height).
  • Cylindrical Vector (P) = a_r * i_r + a_φ * i_φ + a_z * i_z

Visualization

• Top view of the point in the cylindrical system.
• Understanding the representation of the cylinder:
  • XY plane observed from the top.
  • Radius (R) denoted as r.
  • Angle φ measured from the x-axis.
  • Height (Z) along the z-axis.

Directions

• i_r (Radial Direction):
  • Points radially outward from the origin.
• i_φ (Tangential Direction):
  • Tangent to the cylinder's surface from x to y direction.
• i_z (Height Direction):
  • Perpendicular to the XY plane, pointing upwards.

Calculation of Components

Finding a_r (Radial Component)

• Dot Product:
  • a_r = P • i_r
• Using angles between vectors:
  • Angle between x-axis and radial direction (i_r) = φ
  • Angle between y-axis and radial direction = 90° - φ
  • Angle between z-axis and radial direction = 90°
• Formula for a_r:
  • a_r = ax * cos(φ) + a_y * sin(φ)

Finding a_φ (Tangential Component)

• Dot Product:
  • a_φ = P • i_φ
• Using angles:
  • Angle between x-axis and i_φ = 90° + φ
  • Angle between y-axis and i_φ = φ
  • Angle between z-axis and i_φ = 90°
• Formula for a_φ:
  • a_φ = -ax * sin(φ) + a_y * cos(φ)

Finding a_z (Height Component)

• a_z remains the same for both Cartesian and cylindrical vectors.

Example Problem

• Given Cartesian vector components to convert into cylindrical vector.
• Calculate angle φ:
  • φ = tan⁻¹(y/x)
• Determine a_r:
  • Substitute values into a_r formula.
• Determine a_φ:
  • Substitute values into a_φ formula.
• Determine a_z:
  • Directly taken from Cartesian vector.

Conclusion

• Resulting cylindrical vector representation.
• Key formulas to remember:
  • a_r = ax * cos(φ) + a_y * sin(φ)
  • a_φ = -ax * sin(φ) + a_y * cos(φ)
• Importance of practicing the conversion process for exams.

Final Note

• Encouragement to share questions or comments for further help.