Vectors and Scalar Quantities

Jul 16, 2024

Vectors and Scalar Quantities

Definitions

  • Vector Quantity: Has both magnitude and direction (e.g., velocity).
  • Scalar Quantity: Only has magnitude (e.g., speed, temperature, mass, volume).

Examples

  • Speed (Scalar): 40 meters per second.
  • Velocity (Vector): 40 meters per second north.
  • Force (Vector): 300 newtons east, 200 newtons north.
  • Temperature (Scalar): Cannot apply direction.
  • Mass (Scalar): Cannot apply direction.
  • Volume (Scalar): Cannot apply direction.

Visual Representation

  • Directed Line Segment: Initial point to terminal point.
    • Example: Vector AB indicated with an arrow.
    • Magnitude: Length of the vector.
    • Direction: Indicated by the arrow.

Describing Vectors

  • Using Magnitude and Angle:
    • Length (e.g., 5) and angle (e.g., 40 degrees).
  • Using Components:
    • Example: Vector A (2, 3) where 2 is the x-component and 3 is the y-component.

Points vs. Vectors

  • Point Representation: (x, y) in parentheses.
  • Vector Representation: <x, y> in inequality symbols.

Example Problem

  1. Find Component Form: Given initial point (1, -2) and terminal point (5, 1).
    • Calculate x-component: 5 - 1 = 4
    • Calculate y-component: 1 - (-2) = 3
    • Component form: <4, 3>
  2. Calculate Magnitude:
    • Formula: sqrt(x^2 + y^2)
    • Example: sqrt(4^2 + 3^2) = 5
  3. Vector Equivalence: Same magnitude and direction

Operations with Vectors

  • Addition: Connecting vectors head to tail.
    • Example: Vector C = A + B
  • Subtraction: Adding negative of a vector.
    • Example: Vector D = B - A

Position Vectors

  • Initial point at origin.
    • Example: Vector V = <3, 4>

Unit Vectors

  • Magnitude of 1.
    • Formula: Vector v / |v|
  • Standard Unit Vectors:
    • i (x-component), j (y-component), k (z-component)
    • Example: V = 4i - 7j

Conversion between Angles and Components

  • Unit Circle Representation:
    • Cosine for x-component
    • Sine for y-component
  • Formula: V = |V|[cos(θ)i + sin(θ)j]

Solving Problems

  1. Find Magnitude and Angle: For given vector components.
  2. Add Two Vectors: Calculate resultant vector.

Practical Problems

  • Practice: 2A + 3B and 5A - 4B.
  • Resultant force calculation: Magnitude and direction determination.

Conclusion

  • Vectors include both magnitude and direction while scalars only include magnitude.
  • Understanding and representing vectors graphically and component-wise is crucial for solving related problems.