Overview
This lecture covers coordinate systems (rectangular and polar), plotting points in both forms, and the conversion between rectangular and polar coordinates. It also discusses complex numbers, their polar forms, key operations, and includes detailed worked examples and proofs.
Rectangular & Polar Coordinate Systems
- The rectangular (Cartesian) coordinate system divides the plane into four quadrants.
- In the first quadrant, both x and y are positive; in the second, x is negative, y is positive; in the third, both are negative; in the fourth, x is positive, y is negative.
- The origin is the point (0, 0); in polar coordinates, it's called the pole.
- The polar coordinate system represents points as (r, θ), with r as radius and θ as angle from the polar axis.
- Angles are measured as 0° at the positive x-axis, 90° at positive y, 180° at negative x, 270° at negative y, and 360° completes the circle.
Plotting Points in Polar Coordinates
- To plot a point (r, θ), draw a circle of radius r and mark the position at angle θ.
- If r is negative, plot in the direction opposite θ.
- Example: (2, 75°) means two units from the pole at 75°.
- For fractional or negative r-values, adjust the number of circles and direction accordingly.
Conversion Between Rectangular and Polar Form
- To convert (x, y) to (r, θ):
- r = √(x² + y²)
- θ = tan⁻¹(y/x), adjust θ based on the quadrant where (x, y) lies.
- For (r, θ) to (x, y):
Complex Numbers and Polar Form
- The complex number z = x + iy can be written in polar form: z = r(cos θ + i sin θ) or r cis θ.
- r = |z| = √(x² + y²)
- θ (the argument) = tan⁻¹(y/x), taking quadrant into account.
- Polar forms simplify multiplication and division:
- Multiplication: z₁z₂ = r₁r₂ cis(θ₁ + θ₂)
- Division: z₁/z₂ = (r₁/r₂) cis(θ₁ − θ₂)
Operations & Examples with Complex Numbers
- Polar and rectangular forms can be converted for calculations.
- Addition and subtraction are performed in rectangular form; multiplication and division are easier in polar form.
- Argument formulas:
- Argument(z₁z₂) = argument(z₁) + argument(z₂)
- Argument(z₁/z₂) = argument(z₁) − argument(z₂)
- Modulus (magnitude): |z| = √(x² + y²)
- Worked examples illustrate each conversion and operation.
Key Terms & Definitions
- Cartesian/Rectangular Coordinates — (x, y) values on a grid
- Polar Coordinates — (r, θ), with r = distance from pole, θ = angle
- Origin/Pole — Central reference point (0, 0)
- Complex Number — Number in form z = x + iy
- Modulus — Magnitude of complex number, |z| = √(x² + y²)
- Argument — Angle θ of complex number, θ = tan⁻¹(y/x)
- cis θ — Short for cos θ + i sin θ
Action Items / Next Steps
- Practice plotting given points in polar and rectangular systems.
- Convert given complex numbers to polar form and vice versa.
- Complete all parts of Exercise 1.5 and associated homework.
- Purchase or review the recommended textbook for further examples.