Lecture Notes: Introduction to Vectors

Key Concepts

• Vectors vs Scalars
  • Vector: Has both magnitude and direction.
    • Examples: Displacement, Velocity, Acceleration, Force
  • Scalar: Has only magnitude, no direction.
    • Examples: Distance, Speed, Mass

Definitions

• Displacement: Distance with direction.
  • Example: "A person ran 45 meters east" (displacement) vs. "A person ran 45 meters" (distance).
• Velocity: Speed with direction.
  • Example: "A car is going 60 km/h north".
• Acceleration: Change in velocity over time, also a vector.
• Mass: Scalar quantity, e.g., "An object has a mass of 10 kilograms" (no direction).

Problem-Solving: Force Vector Components

• A force vector at an angle has both x and y components.
• Equations for Components
  • Y Component:
    • [ F_y = F \cdot ext{sine} \theta ]
  • X Component:
    • [ F_x = F \cdot ext{cosine} \theta ]
  • Magnitude of Vector:
    • [ F = \sqrt{F_x^2 + F_y^2} ]
  • Angle:
    • [ \theta = \text{arctan}\left(\frac{F_y}{F_x}\right) ]

Example Problem

• Given:
  • Force ( F = 100 \text{ N} ) at an angle of 30 degrees above the x-axis.
• Calculate x and y components:
  • X Component:
    • [ F_x = 100 \cdot \text{cos}(30) = 100 \cdot \frac{\sqrt{3}}{2} \approx 86.6 \text{ N} ]
  • Y Component:
    • [ F_y = 100 \cdot \text{sine}(30) = 100 \cdot \frac{1}{2} = 50 \text{ N} ]
• Expressing in standard unit vectors (i, j):
  • [ F = 86.6 \hat{i} + 50 \hat{j} ]

Summary

• Understanding vectors and their components is crucial for solving problems in physics.
• Pay attention to the distinction between scalar and vector quantities, particularly in various contexts.