Overview
This lecture covers the fundamentals of vectors, including their definitions, differences from scalars, vector representation, types of vectors (equal, parallel, collinear), magnitude calculation, and vector addition and subtraction, with worked examples.
Scalars vs. Vectors
- Scalar quantities have magnitude only (e.g., height, area, speed, distance).
- Vector quantities have both magnitude and direction (e.g., force, velocity, displacement).
- Speed is scalar; velocity is vector because direction matters.
- Distance is scalar; displacement is vector (shortest path between two points).
Basics of Vectors
- A vector from point A to B is written as AB with an arrow indicating direction.
- Reversing direction (from B to A) makes the vector negative: BA = –AB.
- Equal vectors have exactly the same direction and magnitude.
- Parallel vectors share direction but can differ in magnitude: WX = kYZ, where k is a constant.
- Collinear vectors are parallel vectors lying on the same straight line and sharing a point.
- Magnitude (length) of a vector is denoted by modulus bars: |a|.
Calculating Magnitude & Direction
- Use Pythagoras’ theorem: For vectors at right angles, |a| = √(a₁² + a₂²).
- To find direction (bearing), use trigonometric ratios (e.g., tan⁻¹(opposite/adjacent)).
- Bearings are usually given as three-digit numbers indicating the angle from North.
Types of Vector Relationships
- Equal vectors: same magnitude and direction.
- Parallel vectors: can be expressed as multiples of each other.
- Collinear vectors: parallel, on the same line, and share a point.
Addition & Subtraction of Vectors
- For parallel vectors: add or subtract magnitudes directly.
- For non-parallel vectors, use:
- Triangle Law: AB + BC = AC.
- Parallelogram Law: resultant is the diagonal of the constructed parallelogram.
- Polygon Law: resultant is the vector from the start to endpoint following the connected vectors.
- Direction and magnitude in resultant vectors are often determined using Pythagoras’ theorem and trigonometry.
Worked Examples Summary
- Magnitude and direction: Use grid length for magnitude and bearings for direction.
- Expressing vectors in terms of another vector: Use ratios and constants.
- Proving parallel and collinear: Show proportional relationships and common points.
- Solving for unknown coefficients in vector equations: Set up equations by comparing coefficients.
Key Terms & Definitions
- Scalar — Quantity with magnitude only, no direction.
- Vector — Quantity with both magnitude and direction.
- Magnitude — The length or size of a vector, |a|.
- Parallel Vectors — Vectors with the same direction, possibly different magnitudes.
- Equal Vectors — Vectors with both the same magnitude and same direction.
- Collinear Vectors — Parallel vectors that lie on the same straight line and share a point.
- Resultant Vector — The sum of vectors, giving the overall effect.
Action Items / Next Steps
- Practice questions on vector addition, subtraction, and expressing vectors in terms of others.
- Review definitions and properties of vector types.
- Work on identifying direction and magnitude using trigonometry and Pythagoras’ theorem.
- Prepare for next lecture or assigned homework.