Lecture on Vectors

Introduction to Vectors

• Definition: Vectors are measurements with both magnitude (size) and direction.
• Comparison with Scalars:
  • Scalars only have magnitude (e.g., mass, speed, distance, temperature).
  • Vectors have both magnitude and direction (e.g., weight, force, velocity, displacement, acceleration).
• Representation: Vectors can be represented with arrows.
  • Arrow Direction: Indicates the direction of the vector.
  • Arrow Size: Represents the magnitude of the vector.

Adding Vectors

• Example: Displacement of 10m north + 20m east.
  • Resultant vector is the direct path from start to finish (as the crow flies).
  • If vectors are at right angles, use Pythagorean theorem to find the resultant.
  • Calculate the angle (theta) for resultant vector: ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} ) ( (\tan^{-1} \text{ to find } \theta) ).

Resultant Force Example

• Scenario: A ball is pushed with forces of 3N up, 4N right, and 5N in another direction.
  • Add vectors head to tail.
  • Resultant vector in this closed loop scenario is zero (balanced forces, relates to Newton's first law).

Resolving Vectors

• Concept: Use angles to find component or resultant vectors.
• Example: Ship traveling north with a crosswind.
  • Objective: Find the northern component of velocity.
  • Use ( \cos ) if angle is between resultant and component vectors.
  • Use ( \sin ) if angle is not between, as described by mnemonic "turn away from your sin".
  • Resultant and component vectors: Vertical (VV) and Horizontal (VH) components.

Calculation Strategy

• Cosine and Sine:
  • ( \cos \theta ) for angles between resultant and components.
  • ( \sin \theta ) if 'turning away' from angle.
  • Example calculations provided to find components and resultant.

Applications in Physics

• Work Done: ( E = F \cos(\theta) d )
  • Only the component of the force parallel to the direction of movement does work.
• Moments: Similar principle but requires force and distance to be perpendicular.
  • Example: Seesaw with forces applied at an angle; use cosine to find necessary force.

Conclusion and Further Learning

• Emphasis: Importance of understanding vector resolution for exams.
• Recommendation: Watch tutorial videos and walkthroughs for practice on past questions.

Mnemonics and Tips

• "Turn away from your sin": Use sine when turning away from angle.
• "Turn through angle": Use cosine when angle is between resultant and components.

These notes summarize key concepts of vector operations and applications, useful for understanding physics in both theoretical and practical contexts.