Overview
This lecture focuses on distinguishing between scalar and vector quantities, calculating vector components, and expressing vectors using unit vectors.
Scalar vs Vector Quantities
- Scalar quantities have magnitude only, no direction (e.g., temperature, mass, distance, speed).
- Vector quantities have both magnitude and direction (e.g., force, displacement, velocity, acceleration).
- Mass is a scalar because it has no direction; force is a vector due to its directional aspect.
Vector Components and Trigonometry
- Any vector can be broken into x and y components using trigonometric functions.
- The y component (F_y) is found by F_y = F * sin(θ), where θ is the angle above the x-axis.
- The x component (F_x) is found by F_x = F * cos(θ).
- The angle θ can be calculated by θ = arctan(F_y / F_x).
Calculating Vector Magnitude
- The magnitude of a vector from its components is |F| = √(F_x² + F_y²) (Pythagorean theorem).
Example: Force Vector Calculation
- Given a force of 100 N at 30° above the x-axis:
- F_x = 100 * cos(30°) = 86.6 N
- F_y = 100 * sin(30°) = 50 N
Standard Unit Vectors
- Unit vectors have magnitude 1 and point along coordinate axes.
- i: unit vector in x-direction; j: in y-direction; k: in z-direction.
- The force vector can be written as F = 86.6i + 50j.
Key Terms & Definitions
- Scalar — Quantity with magnitude only, no direction.
- Vector — Quantity with both magnitude and direction.
- Unit Vector — A vector with a magnitude of one, indicating direction along an axis.
- Component — The projection of a vector along the coordinate axes.
- Pythagorean theorem — Relationship for right triangles: c² = a² + b².
Action Items / Next Steps
- Memorize the formulas for vector components and magnitude.
- Practice expressing vectors in both magnitude-angle and component forms.
- Review trigonometric functions (sine, cosine, tangent) and their application to vectors.