Overview
This lecture introduces vectors in physics, differentiates between scalar and vector quantities, explains how to represent, measure, and classify vectors, and discusses direction and magnitude in detail.
Scalar and Vector Quantities
- Scalar quantities have magnitude and units but no direction (e.g., temperature, time).
- Vector quantities have magnitude, units, and direction (e.g., weight, velocity, force, displacement, acceleration, momentum, torque).
- Only vector quantities can be associated with a specific direction.
Representation of Vectors
- Vectors are graphically shown as arrowed lines; the arrow indicates direction, and length is proportional to magnitude.
- Vectors are notated with a capital letter and an arrow above, or a boldface letter.
- The magnitude of a vector is written as a regular letter or the vector symbol inside vertical bars (e.g., |A|).
Determining Vector Direction
- Direction is measured as the acute angle (less than 90°) from the reference axis (usually the east-west/x-axis).
- Directions use "degrees ___ of ___" format (e.g., 60° S of W means 60 degrees south of west).
- North/south is written after the angle, followed by east/west (e.g., 30° N of E).
Graphical Representation & Quadrants
- Vectors are placed on a Cartesian plane (x and y axes), divided into four quadrants, each 90°:
- Quadrant I: 0–90° (N of E)
- Quadrant II: 90–180° (N of W)
- Quadrant III: 180–270° (S of W)
- Quadrant IV: 270–360° (S of E)
- Use a protractor to measure angles from the appropriate axis.
Standard and Reference Angles
- Standard angle: measures direction in a full revolution (0–360°).
- Reference angle: smallest positive acute angle with respect to the x-axis.
- Quadrant formulas for reference angle:
- Quadrant I (<90°): reference angle = standard angle
- Quadrant II (90°–180°): reference angle = 180° – standard angle
- Quadrant III (180°–270°): reference angle = standard angle – 180°
- Quadrant IV (270°–360°): reference angle = 360° – standard angle
Magnitude of a Vector
- Magnitude is the length of the vector, measured in units (e.g., km, N).
- Magnitude can be scaled on diagrams (e.g., 1 cm = 1 km).
- Formula: magnitude represented as |A| or a numerical value with units.
Types of Vectors
- Equal vectors: same magnitude and direction.
- Parallel vectors: same direction, different magnitudes.
- Antiparallel vectors: opposite directions (180° apart).
- Non-collinear vectors: separated by an angle ≠0° or 180°, can be acute (<90°) or obtuse (>90°).
Key Terms & Definitions
- Scalar — Quantity with magnitude and units, no direction.
- Vector — Quantity with magnitude, units, and direction.
- Magnitude — Length or size of a vector.
- Reference Angle — Smallest acute angle a vector makes with the horizontal axis.
- Standard Angle — Full-circle angle (0–360°) from the positive x-axis.
- Cartesian Plane — Graph divided into four quadrants by x and y axes.
- Collinear — Vectors lying on the same line.
- Non-collinear — Vectors not on the same line.
Action Items / Next Steps
- Practice representing vectors graphically on the Cartesian plane.
- Review and memorize quadrant formulas for finding reference angles.
- Prepare for the next lecture on vector addition and vector component form.