Overview
This lecture introduces vectors and scalars in physics, details how to represent and describe vectors, and explains types of vector relationships.
Scalars and Vectors
- Scalar quantities are described by magnitude and units only (e.g., temperature, time).
- Vector quantities have both magnitude, units, and a specific direction (e.g., velocity, force, displacement).
- Examples of vector quantities: weight, velocity, force, displacement, acceleration, momentum, torque.
Representing Vectors
- Vectors are shown as lines with arrowheads; the arrow indicates direction.
- The length of the arrow is proportional to the vector's magnitude.
- Vectors use capital letters with an arrow above (→) or boldface; magnitude is shown as a plain letter or inside vertical bars (|A|).
Determining Direction
- The direction of a vector is the acute angle it makes with the east-west (x-axis) line.
- Directions are described as degrees north/south of east/west, e.g., 60° S of W.
Graphical Representation
- Use a Cartesian plane (x and y axes) to plot vectors and identify quadrants.
- Each quadrant spans 90°, totaling 360° for a full revolution.
- Draw angles from the reference axis using a protractor and scale the line appropriately.
Angles and Reference Angles
- Standard angle: measured from the positive x-axis counterclockwise.
- Reference angle: the smallest positive acute angle between the vector and the horizontal axis.
- Quadrant rules for reference angle:
- Quadrant I (0–90°): reference angle = given angle.
- Quadrant II (90–180°): reference angle = 180° – given angle.
- Quadrant III (180–270°): reference angle = given angle – 180°.
- Quadrant IV (270–360°): reference angle = 360° – given angle.
Magnitude of Vectors
- Magnitude equals the length of the vector and can be measured with a ruler using a chosen scale (e.g., 1 cm = 1 km).
- When provided, state both the magnitude and direction, e.g., 25 km 40° S of W.
Types of Vectors
- Equal vectors: same magnitude and direction.
- Parallel vectors: same direction, possibly different magnitudes.
- Antiparallel vectors: same line, opposite directions (180° apart).
- Non-collinear vectors: not along the same line; separated by angles other than 0° or 180° (e.g., perpendicular).
Key Terms & Definitions
- Scalar — Quantity described by magnitude and units only, no direction.
- Vector — Quantity described by magnitude, units, and direction.
- Magnitude — The size or length of a vector.
- Reference Angle — The smallest positive acute angle from the vector to the x-axis.
- Collinear Vectors — Vectors along the same line of action.
- Non-collinear Vectors — Vectors not along the same line (angle not 0° or 180°).
Action Items / Next Steps
- Review how to represent vectors graphically using Cartesian planes and protractors.
- Prepare for the next lecture on vector addition and component representation.