VECTORS 2D

Vectors in 2D

Defining Direction & Adding Vectors

  • Understanding vectors involves defining their direction and how to add them effectively in a two-dimensional space.

Two-Dimensional Vectors

  • Vectors represent quantities with both magnitude and direction in a two-dimensional plane.

Displacement Example

  • Concept: Walking example

    • If you walked 3 blocks North and then 4 blocks East:

      • The displacement is not simply additive (3 + 4 != 5).

Working with Direction in 2D

  • When adding vectors, the direction influences the resultant.

    • Possible outcomes when combining directions include:

      • Minimum displacement = 1 (4 East + 3 West).

      • Maximum displacement = 7 (3 East + 4 East).

Angle Calculation

  • Triangle Example:

    • In the triangle formed:

      • You can use Pythagorean theorem to calculate the angle θ:

        • Tan θ = Opposite/Adjacent = 4/3

        • θ = Tan⁻¹(4/3) ≈ 53.1°

Orientation of Angles

  • The angle 53.1° must be related back to the starting point.

    • Example:

      • 5 = hypotenuse, TAN θ = 4/3, hence θ ≈ 53.1°

Describing Direction

  • Different methods to describe angles:

    • Mathematics: Angle measured from +x-axis, counterclockwise.

    • Aviation: Bearings - North = 0 and goes clockwise.

    • Physics: Describe it in relation to the originating vector's direction and the resultant's direction.

Angle Representation

  • Various ways to describe angles:

    • Examples:

      • 40° N of W

      • 18° W of S

      • 28° S of E

      • 43° E of S

      • Notated angles, e.g., θB S of W, θA E of N.

Drawing Vectors

  • Create vector diagrams from angles:

    • Examples:

      • 42° E of N

      • 31° S of W

      • Others provided in class exercise.

Adding Vectors Not on Axes

  • For vectors not aligned with standard directions:

    • Key Steps:

      1. Separate vectors into their x (horizontal) and y (vertical) components.

      2. Use trig to define components:

        • sin θ = opp/hyp

        • cos θ = adj/hyp

      3. Indicate component directions with + or -.

          • for North & East; - for South & West.

Vector Addition Steps

  1. Break down each vector into x and y components.

  2. Sum all x-components and sum all y-components separately.

  3. Use Pythagorean theorem to determine resultant magnitude.

  4. Calculate angle using tan⁻¹ to find the direction of the resultant vector.

  5. Clearly state the resultant vector's magnitude, unit, and direction.

Example Calculations

  • More complex vectors to add:

    1. Example 1: 14 km North + 18 km @ 30° South of West → Result: 16.3 km @ 17.8° N of W.

    2. Example 2: 12 m/s East + 8.2 m/s @ 11° West of North…

Dynamics

Definition

  • Dynamics is the study of forces and their effect on motion.

Fundamental Forces

  • Types of Forces:

    • Electromagnetic (e.g. photons), Nuclear (strong and weak), Gravitational.

    • Each force has specific characteristics such as range and particle interactions.

Newton's Laws of Motion

1st Law: Inertia

  • An object not acted upon by a force remains at rest or continues moving uniformly.

2nd Law: F = ma

  • Acceleration is produced by net force proportional to that force and inversely proportional to mass.

3rd Law: Action-Reaction

  • For every action, there is an equal and opposite reaction.

Measuring Forces

  • All force measures are in Newtons, defined as the force needed to accelerate 1 kg at 1 m/s².

Example Problems

  1. Calculate net force required to maintain a moving vehicle.

  2. Determine the acceleration of a car subjected to opposing forces.

  3. Analyze a crate being pushed by multiple forces to find its acceleration.