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1.3.1. Cartesian Coordinates

Table of Contents

The Coordinate Grid

Cartesian coordinates describe position by measuring how far a point is from fixed reference lines called axes. In two dimensions, we use two perpendicular axes, the horizontal $x$ axis and the vertical $y$ axis. Their intersection is called the origin, written as $(0,0)$.

A point in the plane is written as an ordered pair $(x,y)$. The first number tells how far to move along the $x$ direction, and the second tells how far to move along the $y$ direction. For example, the point $(3,2)$ means move 3 units to the right and 2 units upward from the origin.

In three dimensions, a third axis is added, called the $z$ axis. A point is then written as $(x,y,z)$. This lets us describe position in space, not just on a flat surface.

Cartesian axes in two dimensions

Reading Coordinates

The order of the numbers matters. The point $(2,5)$ is not the same as $(5,2)$. In Cartesian coordinates, each coordinate belongs to a specific axis.

In physics, Cartesian coordinates are useful because each direction is treated independently. A particle may move in the $x$ direction, the $y$ direction, and the $z$ direction, and we can study each part of the motion separately.

A Cartesian position must be written in the correct order.
In 2D: $(x,y)$
In 3D: $(x,y,z)$
Changing the order changes the point.

Signs and Direction

Coordinates can be positive, negative, or zero. Positive $x$ usually means right, negative $x$ means left. Positive $y$ usually means up, negative $y$ means down. In three dimensions, positive and negative $z$ indicate opposite directions perpendicular to the $xy$ plane.

This sign system lets Cartesian coordinates describe position relative to the origin very clearly.

Coordinate signMeaning
$x > 0$right of the origin
$x < 0$left of the origin
$y > 0$above the origin
$y < 0$below the origin
$z > 0$positive $z$ direction
$z < 0$negative $z$ direction

Quadrants in Two Dimensions

The two axes divide the plane into four regions called quadrants. These help describe where a point lies.

QuadrantSign of $x$Sign of $y$
I$+$$+$
II$-$$+$
III$-$$-$
IV$+$$-$

For example, the point $(-2,4)$ lies in Quadrant II, while $(3,-1)$ lies in Quadrant IV.

The four quadrants

Cartesian Coordinates in Three Dimensions

In three dimensions, the three axes are all perpendicular to one another. The $x$, $y$, and $z$ coordinates specify a unique point in space. This system is widely used in mechanics, electromagnetism, and many other areas of physics because it matches how we often break motion and force into independent directions.

A point such as $(1,-2,3)$ means 1 unit in the positive $x$ direction, 2 units in the negative $y$ direction, and 3 units in the positive $z$ direction.

Three-dimensional Cartesian axes

Position as Distance from the Origin

A useful quantity in Cartesian coordinates is the straight line distance from the origin to a point. In two dimensions, for a point $(x,y)$, this distance is

$$
r = \sqrt{x^2 + y^2}
$$

In three dimensions, for a point $(x,y,z)$, it becomes

$$
r = \sqrt{x^2 + y^2 + z^2}
$$

This follows from the Pythagorean theorem.

Distance from the origin in Cartesian coordinates:
$$
r = \sqrt{x^2 + y^2}
$$
in 2D, and
$$
r = \sqrt{x^2 + y^2 + z^2}
$$
in 3D.

Why Cartesian Coordinates Are Important in Physics

Cartesian coordinates are often the simplest system when motion or forces happen along straight, perpendicular directions. A falling object moving straight up and down can be described with one coordinate. A ball moving across a floor may need two coordinates. A flying object in space may need three.

This coordinate system is especially convenient when equations separate naturally into $x$, $y$, and $z$ parts. That is one reason it is the standard starting point in physics.

A Simple Example

Suppose a particle is at the point $(4,-3)$. This means it is 4 units to the right of the origin and 3 units below it. Its distance from the origin is

$$
r = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
$$

So the particle is 5 units from the origin.

Summary

Cartesian coordinates locate points using perpendicular axes. In two dimensions, positions are written as $(x,y)$, and in three dimensions as $(x,y,z)$. The sign of each coordinate tells direction relative to the origin, and the system is especially useful in physics because it lets us describe each spatial direction clearly and separately.

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