Overview
This lecture introduces vectors, covering the distinction between scalars and vectors, trigonometry basics for vector components, and practical methods for adding, subtracting, and multiplying vectors in two dimensions.
Trigonometry Review for Vectors
- SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- The Pythagorean theorem for right triangles: a² + b² = c², c is the hypotenuse.
- Use inverse tangent to find angles: θ = tan⁻¹(opposite/adjacent).
- Sine and cosine functions let you find triangle sides if you know the hypotenuse and angle.
Scalars vs. Vectors
- Scalar: only magnitude (e.g., speed, energy).
- Vector: magnitude and direction (e.g., velocity, displacement, force).
- Velocity is a vector; speed is a scalar.
Vector Notation and Representation
- Vectors are drawn as arrows on Cartesian planes; arrow direction = vector direction, length = magnitude.
- Use an arrow above a variable to denote it as a vector.
- Vector direction is typically given relative to the positive x-axis.
Breaking Vectors into Components
- Any vector can be expressed as its x and y components using a right triangle.
- x-component: Ax = |A| cos(θ) where θ is with respect to the x-axis.
- y-component: Ay = |A| sin(θ).
- Always prefer the angle with respect to the positive x-axis for convenience and sign correctness.
Calculating Vector Components (Examples)
- For a vector of 18 units at 60°: x = 18 cos(60°) = 9, y = 18 sin(60°) = 15.6.
- Quadrant affects sign: second quadrant x-component is negative, third quadrant x and y are negative.
- Use the proper angle (e.g., 120° for second quadrant) to ensure correct signs with sine/cosine.
Adding and Subtracting Vectors
- To add vectors: break each into components, sum all x-components, sum all y-components.
- Resultant vector magnitude: |R| = √(Rx² + Ry²), direction: θ = tan⁻¹(Ry/Rx).
- Tip-to-tail method: placing vectors sequentially to find resultant.
- To subtract, add the negative of a vector (reverse direction or subtract components).
Multiplying Vectors by Scalars
- Multiplying a vector by a scalar changes its magnitude only, not direction.
- All components are multiplied by the scalar.
Key Terms & Definitions
- Vector — A quantity with both magnitude and direction.
- Scalar — A quantity with magnitude only.
- Component — The projection of a vector along the x or y-axis.
- Resultant Vector — The vector sum of two or more vectors.
- Pythagorean Theorem — In a right triangle, a² + b² = c².
- SOHCAHTOA — Mnemonic for sine, cosine, and tangent trigonometric ratios.
Action Items / Next Steps
- Practice breaking vectors into x and y components.
- Practice assembling components back into resultant vectors.
- Practice adding and subtracting vectors by their components.
- Prepare for upcoming homework on one-dimensional and two-dimensional vector problems.