Lecture Notes on Vectors

Key Concepts

• Vector vs Scalar Quantities
  • Scalar Quantity: Has magnitude but no direction.
    • Example: Temperature (e.g., 80°F)
  • Vector Quantity: Has both magnitude and direction.
    • Example: Force (e.g., 100 N at 30 degrees above the x-axis)

Important Definitions

• Displacement vs Distance

  • Distance: Scalar quantity (e.g., 45 meters).
  • Displacement: Vector quantity; distance with direction (e.g., 45 meters east).

• Speed vs Velocity

  • Speed: Scalar quantity (how fast).
  • Velocity: Vector quantity; speed with direction.

• Acceleration:

  • Also a vector; measures how fast velocity changes.

• Mass:

  • Scalar quantity (e.g., 10 kg) with no direction.

Problem Solving with Vectors

Force Vector Example

• Given:
  • Force vector magnitude: 100 N
  • Angle: 30 degrees above the x-axis

Component Calculation

• Breaking Down Force Vector (F):
  • X Component (F_x):
    • Formula: F_x = F * cos(theta)
  • Y Component (F_y):
    • Formula: F_y = F * sin(theta)

Trigonometric Relationships

• SOHCAHTOA
  • Sine: sin(theta) = opposite/hypotenuse
  • Cosine: cos(theta) = adjacent/hypotenuse
  • Tangent: tan(theta) = opposite/adjacent

Pythagorean Theorem

• Relationship: c² = a² + b²
  • Where c = hypotenuse (F), a = F_x, b = F_y
  • Magnitude of a vector: |F| = √(F_x² + F_y²)

Example Calculation

• X Component Calculation:

  • F_x = 100 * cos(30°)
  • cos(30°) = √3/2
  • F_x = 100 * (√3/2) = 86.6 N

• Y Component Calculation:

  • F_y = 100 * sin(30°)
  • sin(30°) = 1/2
  • F_y = 100 * (1/2) = 50 N

Unit Vectors

• Definition: A vector with a magnitude of one.
• Standard Unit Vectors:
  • i: Unit vector along the x-axis
  • j: Unit vector along the y-axis
  • k: Unit vector along the z-axis

Final Expression of Force Vector

• Force vector in component form:
  • F = 86.6i + 50j (in Newtons)

Summary

• Understanding vectors involves both magnitude and direction.
• Key formulas include those for breaking down force vectors and expressing them using unit vectors.
• Practice with examples is crucial for mastering vector calculations.