Polar Coordinates and Vectors

Flashcards for reviewing key concepts related to polar coordinates and vectors.

Polar Coordinates

A system for representing points in a plane using the distance from a reference point (pole) and the angle from a reference direction (polar axis).

Pole

The central point in the polar coordinate system from which distances are measured.

r

The distance from the pole to a point in polar coordinates.

θ (theta)

The angle formed by the polar axis and the line from the pole through the given point, measured in degrees or radians.

Negative r

When r is negative in polar coordinates, it indicates a point in the opposite direction along the ray defined by the angle θ.

Rectangular Coordinates

A coordinate system that defines a point by its distance from two perpendicular axes, usually labeled x and y.

Conversion from polar to rectangular coordinates

The process of translating polar coordinates (r, θ) into rectangular coordinates (x, y) using the formulas x = r cos(θ) and y = r sin(θ).

Conversion from rectangular to polar coordinates

The process of translating rectangular coordinates (x, y) into polar coordinates (r, θ) using the formulas r = √(x² + y²) and θ = tan⁻¹(y/x).

Polar Graphs

Graphs that represent equations in polar coordinates, often requiring conversion to rectangular form for graphing.

Limaçon

A type of polar graph that has a distinctive shape; can have an inner loop or be smooth depending on parameters.

Cardioid

A special case of a limaçon with a single cusp that resembles a heart shape, represented by equations of the form r = a(1 + cos(θ)) or r = a(1 - cos(θ)).

Dot Product

A scalar product of two vectors, calculated as the sum of the products of their corresponding components, often used to determine angle relationships between vectors.

Cross Product

A vector operation on two vectors in three-dimensional space that results in another vector that is orthogonal to both of the original vectors.

Work

The product of force and distance, and is calculated as W = F imes d imes cos(θ) where θ is the angle between the force and the direction of motion.