Overview
This lecture introduces scalar and vector quantities, explains key differences, and demonstrates how to resolve force vectors into components using trigonometry, including expressing vectors with unit vectors.
Scalars vs. Vectors
- Scalars have magnitude only, no direction (e.g., temperature, mass, distance, speed).
- Vectors have both magnitude and direction (e.g., force, displacement, velocity, acceleration).
- Displacement = distance with direction; velocity = speed with direction.
- Mass is a scalar, not a vector.
Resolving Force Vectors into Components
- Any vector can be broken into x (horizontal) and y (vertical) components.
- For a force vector F at angle θ:
- X-component: ( F_x = F \cos \theta )
- Y-component: ( F_y = F \sin \theta )
- Example: F = 100 N, θ = 30°
- ( F_x = 100 \cos 30^\circ = 86.6 ) N
- ( F_y = 100 \sin 30^\circ = 50 ) N
Trigonometric Relationships & Formulas
- Sine: ( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} )
- Cosine: ( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} )
- Tangent: ( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} )
- To find angle: ( \theta = \arctan \left( \frac{F_y}{F_x} \right) )
- Pythagorean theorem for vector magnitude: ( F = \sqrt{F_x^2 + F_y^2} )
Unit Vectors and Standard Form
- A unit vector has magnitude 1; used to indicate direction along axes.
- ( \mathbf{i} ) = unit vector in x-direction, ( \mathbf{j} ) = y-direction, ( \mathbf{k} ) = z-direction.
- Express force vector: ( \mathbf{F} = 86.6 \mathbf{i} + 50 \mathbf{j} ) (N).
Key Terms & Definitions
- Scalar — quantity with magnitude only.
- Vector — quantity with both magnitude and direction.
- Unit Vector — vector with magnitude of 1 along an axis (( \mathbf{i}, \mathbf{j}, \mathbf{k} )).
Action Items / Next Steps
- Memorize scalar vs. vector examples and key formulas for components and vector magnitude.
- Practice resolving vectors into components and expressing them with unit vectors.