Vectors & Polar Coordinates

Rectangular to Polar Conversion

Converting points and equations between rectangular and polar form.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Vector Operations

Coordinates Built from Distance and Direction

A vector ⟨3, 4⟩ has a magnitude of 5 and points in a specific direction. Before reading on, think about this: instead of describing a location by how far right and how far up (rectangular coordinates), could a point instead be described directly by its distance from the origin and the angle it makes — the same two pieces of information a vector already carries?

Definition — Polar Coordinates

A point described as (r, θ): r is the distance from the origin, and θ is the angle measured counterclockwise from the positive x-axis. Conversions: x = r cos θ, y = r sin θ, r = √(x² + y²), and θ = arctan(y/x), adjusted for the correct quadrant.

These conversions come directly from the unit circle, scaled by r: (cos θ, sin θ) gives the direction on a circle of radius 1, and multiplying both coordinates by r stretches that same direction out to the actual distance r — exactly x = r cos θ and y = r sin θ. The reverse formulas, r = √(x² + y²) and θ = arctan(y/x), are the same magnitude and direction ideas already used for vectors, just renamed.

Worked Example — Converting Rectangular to Polar

Convert (3, 4) to polar coordinates. r = √(9 + 16) = 5. Since (3, 4) is in Quadrant I, θ = arctan(4/3) directly gives the correct angle, approximately 0.927 radians (about 53.1°). Polar form: (5, 0.927).

Worked Example — Converting Polar to Rectangular

Convert (4, π/3) to rectangular coordinates. x = 4cos(π/3) = 4(1/2) = 2. y = 4sin(π/3) = 4(√3/2) = 2√3. Rectangular form: (2, 2√3).

Worked Example — Converting an Equation, Not Just a Point

Convert x² + y² = 9 to a polar equation. Recognize x² + y² is exactly r² (from the conversion formulas): r² = 9, so r = 3 — a circle of radius 3 centered at the origin, described by a dramatically simpler equation in polar form than in rectangular form.

Tip

When finding θ using arctan(y/x), always double-check the point's actual quadrant — arctan alone only returns angles in a limited range and can give a result that's off by π from the true angle for points in Quadrants II and III.

Common Mistakes

Using arctan(y/x) directly without checking the quadrant, such as computing θ = arctan(1) = π/4 for the point (−1, −1), when the true angle is actually in Quadrant III.
arctan only returns angles between −π/2 and π/2 — for a point in Quadrant II or III, add π to the raw arctan result to land in the correct quadrant; for (−1, −1), that gives π/4 + π = 5π/4.
Forgetting that a single rectangular point corresponds to infinitely many equivalent polar representations, by adding any multiple of 2π to θ.
Polar coordinates aren't unique the way rectangular coordinates are — (r, θ) and (r, θ + 2π) describe the exact same point, which is fine as long as one consistent representation is used within a single problem.

Key Takeaways

Polar coordinates (r, θ) describe a point by distance from the origin and angle, the same two quantities a vector's magnitude and direction already capture.
The conversion formulas come from scaling the unit circle's (cos θ, sin θ) by the distance r.
Converting θ using arctan requires checking the point's actual quadrant, since arctan alone doesn't distinguish all four quadrants.

Summary

Polar coordinates reframe location around distance and direction rather than horizontal and vertical displacement. The next lesson graphs equations written directly in this new coordinate system.

Track your progress

← Vector Operations Back to Vectors & Polar Coordinates Graphing Polar Equations→