Understanding Vectors and Their Operations

Oct 11, 2024

Notes on Vectors

Introduction to Vectors

  • Vectors: A physical quantity that has both magnitude and direction.
  • Scalar: A physical quantity that has only magnitude.
    • Examples of scalars: distance, mass, time, speed.
    • Examples of vectors: force, momentum, acceleration, velocity, displacement.

Key Definitions

  • Magnitude: Size of the vector.
  • Direction: The way in which the vector is pointing.

Scalar vs Vector

  • Vector: Has both magnitude and direction (e.g., 20 km/h south).
  • Scalar: Only has magnitude (e.g., 20 km/h).

Vector Operations

  • Vectors can be added together.
    • Example: For vector A (20N) and vector B (30N), the resultant is A + B = 50N if they are parallel.
  • Resultant vector: The sum of two or more vectors.
  • Parallel vectors: Can be added directly.
  • Opposite direction vectors: One vector is negative.

Perpendicular Vectors

  • When adding perpendicular vectors, use the Pythagorean theorem:
    • Resultant = √(A² + B²)
  • Example: A = 20N, B = 10N, Resultant = √(20² + 10²) = 22.36N.

Vectors Not in Standard Direction

  • Vectors not aligned with x or y axes have both x and y components.
  • Resolve vectors into components using trigonometric functions:
    • Ax = A cos(θ)
    • Ay = A sin(θ)
    • Use the angle from the positive x-axis.

Angle Calculations

  • Tan function: To find angles when components are known.
    • tan(θ) = Ay/Ax
    • θ = tan⁻¹(Ay/Ax)

Resolving Vectors

  1. Identify the angle: Measure from the positive x-axis.
  2. Calculate components:
    • Ax = A cos(θ)
    • Ay = A sin(θ)
  3. Sum components to find the resultant.

Sketching Vectors

  • Draw the initial point, then draw the vector to scale according to its magnitude and direction.
  • Each vector’s direction can be indicated with arrows.

Quadrants and Angles

  • First Quadrant: (+x, +y)
  • Second Quadrant: (-x, +y) - add 180 to the angle found.
  • Third Quadrant: (-x, -y) - add 180 to the angle found.
  • Fourth Quadrant: (+x, -y) - add 360 to the angle found.

Example Problems

  • Displacement of a ship: Analyze the components of motion in different directions (e.g., north, east, south)
  • Free-body diagrams: Differentiate between sketching vectors and free body diagrams, where the latter focuses on forces acting on an object.
    • Free body diagrams include all forces acting on an object, while sketches show the resultant path.

Practice Questions

  1. Finding X and Y components of a vector given magnitude and angle.
  2. Determining resultant of multiple vectors.
  3. Understanding vector addition and subtraction concepts based on direction and magnitude.

Conclusion

  • Mastering vectors involves understanding their properties, operations, and how to resolve and sketch them accurately.