Lecture Notes on Vector Formulas in Physics

Introduction

• Discussion on vector formulas relevant to physics
• Vector v with components:
  • X component: ( V_{x} )
  • Y component: ( V_{y} )
• Representation of vectors:
  • Arrow on top indicates a vector
  • Absolute value signs indicate magnitude

Vector Quantities

• Scalar quantity: Magnitude only, no direction
• Vector quantity: Magnitude and direction

Components of Vectors

• 2D Vector Components:
  • ( V_{x} = V \cos \theta )
  • ( V_{y} = V \sin \theta )
  • Magnitude ( |V| = \sqrt{V_{x}^2 + V_{y}^2} ) (Pythagorean theorem)
• 3D Vector Components:
  • ( |V| = \sqrt{V_{x}^2 + V_{y}^2 + V_{z}^2} )

Angle and Direction

• Angle ( \theta ) to find direction of vector:
  • ( \theta = \text{arctan}(\frac{V_{y}}{V_{x}}) )

Representation Using Unit Vectors

• 2D Vectors: ( V = V_{x}i + V_{y}j )
• 3D Vectors: ( V = V_{x}i + V_{y}j + V_{z}k )
• Standard Unit Vectors:
  • ( i = (1, 0, 0) )
  • ( j = (0, 1, 0) )
  • ( k = (0, 0, 1) )

Vector Multiplication

• Dot Product:
  • Result is scalar
  • Formula: ( A \cdot B = |A||B|\cos\theta )
  • Example: Work (Force ( \cdot ) Displacement)
• Cross Product:
  • Result is vector
  • Formula: ( A \times B = |A||B|\sin\theta )
  • Related to determinant of a 3x3 matrix
  • Example: Torque and Magnetic Force

Calculations

• Work: ( W = F \cdot d ), zero when force and displacement are perpendicular
• Torque: ( \tau = r \times F )
• Magnetic Force on Moving Charge: ( F = Q(V \times B) )

Unit Vectors and Position

• Unit Vector: ( R_{hat} = \frac{R}{|R|} )
  • Gives direction
• Position Vector: ( R = (x_b - x_a)i + (y_b - y_a)j + (z_b - z_a)k )

Additional Topics

• Electric Field Vector:
  • Formula: ( E = \frac{kQ}{r^2} R_{hat} )
  • Describes field emitted by a charge

Conclusion

• More formulas available in downloadable formula sheet
• Video resources available for further understanding