Overview
This lecture covers the distinction between scalar and vector quantities, how to resolve vectors into components, and how to express vectors using unit vectors.
Scalars vs Vectors
- Scalar quantities have magnitude only and no direction (e.g., mass, temperature, distance, speed).
- Vector quantities have both magnitude and direction (e.g., force, displacement, velocity, acceleration).
- Displacement is distance with direction, while velocity is speed with direction.
- Mass is a scalar since direction does not apply to it.
Vector Components & Trigonometry
- Any vector can be broken into x and y components using trigonometric functions.
- The y-component: ( F_y = F \cdot \sin(\theta) ), where θ is the angle from the x-axis.
- The x-component: ( F_x = F \cdot \cos(\theta) ).
- To find the angle from components: ( \theta = \arctan(F_y / F_x) ).
- The vector magnitude given components: ( F = \sqrt{F_x^2 + F_y^2} ).
Example: Force Vector Components
- For a force of 100 N at 30° above the x-axis:
- ( F_x = 100 \cdot \cos(30^\circ) = 86.6,\text{N} )
- ( F_y = 100 \cdot \sin(30^\circ) = 50,\text{N} )
Unit Vectors & Component Form
- Unit vectors have a magnitude of 1 and indicate direction: ( \mathbf{i} ) for x, ( \mathbf{j} ) for y, ( \mathbf{k} ) for z.
- Expressing the force vector using unit vectors: ( \mathbf{F} = 86.6,\mathbf{i} + 50,\mathbf{j} ).
Key Terms & Definitions
- Scalar — A quantity with magnitude only, no direction.
- Vector — A quantity with both magnitude and direction.
- Unit vector — A vector of magnitude 1 pointing along an axis (( \mathbf{i} ), ( \mathbf{j} ), or ( \mathbf{k} )).
- Component form — Representation of a vector as the sum of its x, y, and (if 3D) z parts.
Action Items / Next Steps
- Memorize key component formulas for x and y.
- Practice expressing vectors in unit vector form.
- Review the distinction between scalar and vector quantities.