Overview
This lecture covers vector representation in Cartesian (rectangular) coordinates, including position, velocity, and acceleration, with detailed examples of calculating vector components and magnitudes.
Vector Representation in Cartesian Coordinates
- Vectors can be expressed in any coordinate system, but rectangular (XYZ) components are often most convenient.
- A position vector, r, is written as: r = x·î + y·ĵ + z·k̂, where î, ĵ, and k̂ are unit vectors in the X, Y, and Z directions.
- The magnitude of r is |r| = √(x² + y² + z²).
- The direction of r can be represented by the unit vector u_r = r / |r|.
Velocity in Cartesian Coordinates
- Velocity vector v is the time derivative of position: v = dr/dt.
- In component form: v = (dx/dt)·î + (dy/dt)·ĵ + (dz/dt)·k̂, or v_x·î + v_y·ĵ + v_z·k̂.
- Velocity components are: v_x = ẋ, v_y = ẏ, v_z = ż (dots indicate derivatives with respect to time).
- The magnitude of velocity: |v| = √(v_x² + v_y² + v_z²).
- The velocity direction is v / |v| and is always tangent to the path.
Acceleration in Cartesian Coordinates
- Acceleration a is the time derivative of velocity: a = dv/dt = d²r/dt².
- In component form: a = a_x·î + a_y·ĵ + a_z·k̂.
- a_x = dv_x/dt = ẍ; a_y = dv_y/dt = ÿ; a_z = dv_z/dt = z̈ (double dots for second derivatives).
- The magnitude of acceleration: |a| = √(a_x² + a_y² + a_z²).
- Components (a_x, a_y, a_z) are scalars, not vectors.
Example 1: Box Sliding Down a Ramp
- Given path: y = 0.05x²; v_x = -3 m/s; a_x = -1.5 m/s² at x = 5 m.
- To find y-components:
- v_y = d(y)/dt = 2·0.05·x·ẋ; at x = 5, ẋ = v_x = -3 ⇒ v_y = 1.5 m/s.
- a_y = d(v_y)/dt; apply product rule: a_y = 0.1·(ẋ)² + 0.1·x·ẍ; substitute x = 5, ẋ = -3, ẍ = a_x = -1.5 ⇒ a_y = 0.15 m/s².
Example 2: Particle Path and Acceleration Magnitude
- Path: y = 0.5x²; v_x = 5t ft/s; at t = 1 s.
- X position: integrate v_x = 5t ⇒ x = 2.5 ft.
- Y position: y = 0.5·x² = 3.125 ft.
- a_x = d(v_x)/dt = 5 ft/s².
- v_y = d(y)/dt = 2·0.5·x·ẋ; at t = 1, x = 2.5, ẋ = 5 ⇒ v_y = 12.5 ft/s.
- a_y = d(v_y)/dt = ẋ² + x·a_x; plug in ẋ = 5, x = 2.5, a_x = 5 ⇒ a_y = 37.5 ft/s².
- Position magnitude: √(x² + y²) = 4 ft.
- Acceleration magnitude: √(a_x² + a_y²) = 37.8 ft/s².
Key Terms & Definitions
- Vector — A quantity with both magnitude and direction.
- Unit Vector (î, ĵ, k̂) — Vectors of length 1 pointing along the X, Y, or Z axis.
- Magnitude — The length or size of a vector.
- Component — The projection of a vector along an axis.
- Dot Notation — A dot above a variable denotes differentiation with respect to time (e.g., ẋ is dx/dt).
- Product Rule — A calculus rule for differentiating products of two functions.
Action Items / Next Steps
- Practice expressing vectors and calculating their magnitudes and components in Cartesian coordinates.
- Review and solve similar example problems for position, velocity, and acceleration.
- Be comfortable applying chain and product rules for derivatives in physics problems.