Lecture Notes: Vectors

Introduction to Vectors

• Vectors are quantities that have both magnitude and direction.
• Scalars are quantities with magnitude only, no direction.
• Examples of scalar quantities:
  • Temperature
  • Mass
• Examples of vector quantities:
  • Displacement
  • Velocity
  • Acceleration
  • Force

Scalar vs Vector

• Scalar quantities
  • Only magnitude
  • Example: Mass (10 kg), Temperature (80°F)
• Vector quantities
  • Magnitude and direction
  • Example: Force (100 N at 30° above x-axis)

Key Concepts

• Displacement vs Distance
  • Distance: Scalar (e.g., 45 meters)
  • Displacement: Vector (e.g., 45 meters east)
• Speed vs Velocity
  • Speed: Scalar (how fast)
  • Velocity: Vector (how fast and direction)
• Acceleration
  • Vector (rate of change of velocity)

Problem Solving with Vectors

• Force Vector Problem
  • Given: Force vector 100 N at 30° above x-axis
  • Objective: Calculate magnitude of x and y components

Trigonometry Basics

• Use SOHCAHTOA to resolve vectors
  • Sine (SOH): Opposite/Hypotenuse (Fy/F)
  • Cosine (CAH): Adjacent/Hypotenuse (Fx/F)
  • Tangent (TOA): Opposite/Adjacent (Fy/Fx)

Calculating Components

• Y Component (Fy)
  • Formula: Fy = F * sin((θ))
  • Calculation: 100 N * sin(30°) = 50 N
• X Component (Fx)
  • Formula: Fx = F * cos((θ))
  • Calculation: 100 N * cos(30°) = 50√3 ≈ 86.6 N

Expressing Vectors

• Unit Vectors
  • I (x-axis), J (y-axis), K (z-axis)
• Standard Unit Vector Form
  • F = 86.6i + 50j
  • i, j, k represent directions x, y, z respectively

Important Formulas

• Magnitude of a Vector
  • Formula: ( C^2 = A^2 + B^2 ) or ( F = \sqrt{Fx^2 + Fy^2} )
• Angle Calculation
  • Formula: ( \theta = \tan^{-1}(Fy/Fx) )

Conclusion

• Vectors can be described in terms of magnitude & direction or component form.
• Understanding vector components and using trigonometry are crucial for solving physics problems.