Cartesian Coordinates

Cartesian Coordinates
Rectangular Coordinates

(x, y) or (x, y, z) coordinates.

See also

Key Formula

P = (x,\, y)

Where:

$P$ = A point in the two-dimensional Cartesian plane
$x$ = The horizontal distance from the origin along the x-axis (positive to the right)
$y$ = The vertical distance from the origin along the y-axis (positive upward)

Worked Example

Problem: Plot the point A = (3, −2) on the Cartesian plane and find its distance from the origin.

Step 1: Start at the origin (0, 0). The x-coordinate is 3, so move 3 units to the right along the x-axis.

Step 2: The y-coordinate is −2, so from that position move 2 units downward (negative y-direction). Mark the point A there.

Step 3: Use the distance formula to find how far A is from the origin O = (0, 0).

d = \sqrt{(3-0)^2 + (-2-0)^2} = \sqrt{9 + 4} = \sqrt{13}

Answer: Point A is located 3 units right and 2 units below the origin, at a distance of

\sqrt{13} \approx 3.61

units from the origin.

Another Example

Problem: Identify which quadrant the point B = (−5, 4) lies in.

Step 1: Check the sign of the x-coordinate: x = −5, which is negative, so the point is to the left of the y-axis.

Step 2: Check the sign of the y-coordinate: y = 4, which is positive, so the point is above the x-axis.

Step 3: A point with a negative x-coordinate and a positive y-coordinate lies in Quadrant II.

Answer: B = (−5, 4) is in Quadrant II.

Frequently Asked Questions

Why is it called the Cartesian coordinate system?

It is named after the French mathematician René Descartes (Latinized as Cartesius), who in the 17th century developed the idea of describing geometry using algebra. His key insight was that every point in a plane can be uniquely identified by a pair of numbers measured along perpendicular axes.

What are the four quadrants of the Cartesian plane?

The two axes divide the plane into four regions. Quadrant I has (+x, +y), Quadrant II has (−x, +y), Quadrant III has (−x, −y), and Quadrant IV has (+x, −y). Quadrants are numbered counterclockwise starting from the upper-right.

Cartesian Coordinates vs. Polar Coordinates

Cartesian coordinates locate a point using horizontal and vertical distances

(x, y)

from the origin. Polar coordinates instead use a distance

r

from the origin and an angle

\theta

measured from the positive x-axis, written as

(r, \theta)

. You can convert between them:

x = r\cos\theta

and

y = r\sin\theta

. Cartesian coordinates are usually simpler for straight lines and rectangles, while polar coordinates are often more natural for circles and spirals.

Why It Matters

Cartesian coordinates form the foundation of analytic geometry, connecting algebra and geometry so you can graph equations, compute distances, and find slopes. Nearly every graph you encounter in math, science, or engineering is drawn on a Cartesian plane. The same idea extends to three dimensions

(x, y, z)

, which is essential for physics simulations, computer graphics, and GPS-like positioning systems.

Common Mistakes

Mistake: Writing the coordinates in the wrong order, such as (y, x) instead of (x, y).

Correction: The x-coordinate (horizontal) always comes first, and the y-coordinate (vertical) comes second. Think of it alphabetically: x before y.

Mistake: Confusing which quadrant a point belongs to when one or both coordinates are negative.

Correction: Remember the sign pattern by quadrant: I (+, +), II (−, +), III (−, −), IV (+, −). Check each coordinate's sign individually before deciding the quadrant.

Related Terms

Coordinates — General concept of locating points in space
Polar Coordinates — Alternative system using distance and angle
Parametric Equations — Express x and y as functions of a parameter
Ordered Pair — The (x, y) notation used for 2D points
Distance Formula — Calculates distance between two Cartesian points
Midpoint — Finds the point halfway between two coordinates
Quadrant — One of four regions of the Cartesian plane