# A.15: Cartesian Coordinates

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Each point in two dimensions may be labeled by two coordinates $$(x,y)$$ which specify the position of the point in some units with respect to some axes as in the figure below.

The set of all points in two dimensions is denoted $$\mathbb{R}^2\text{.}$$ Observe that

• the distance from the point $$(x,y)$$ to the $$x$$-axis is $$|y|$$
• the distance from the point $$(x,y)$$ to the $$y$$-axis is $$|x|$$
• the distance from the point $$(x,y)$$ to the origin $$(0,0)$$ is $$\sqrt{x^2+y^2}$$

Similarly, each point in three dimensions may be labeled by three coordinates $$(x,y,z)\text{,}$$ as in the two figures below.

The set of all points in three dimensions is denoted $$\mathbb{R}^3\text{.}$$ The plane that contains, for example, the $$x$$- and $$y$$-axes is called the $$xy$$-plane.

• The $$xy$$-plane is the set of all points $$(x,y,z)$$ that obey $$z=0\text{.}$$
• The $$xz$$-plane is the set of all points $$(x,y,z)$$ that obey $$y=0\text{.}$$
• The $$yz$$-plane is the set of all points $$(x,y,z)$$ that obey $$x=0\text{.}$$

More generally,

• The set of all points $$(x,y,z)$$ that obey $$z=c$$ is a plane that is parallel to the $$xy$$-plane and is a distance $$|c|$$ from it. If $$c \gt 0\text{,}$$ the plane $$z=c$$ is above the $$xy$$-plane. If $$c \lt 0\text{,}$$ the plane $$z=c$$ is below the $$xy$$-plane. We say that the plane $$z=c$$ is a signed distance $$c$$ from the $$xy$$-plane.
• The set of all points $$(x,y,z)$$ that obey $$y=b$$ is a plane that is parallel to the $$xz$$-plane and is a signed distance $$b$$ from it.
• The set of all points $$(x,y,z)$$ that obey $$x=a$$ is a plane that is parallel to the $$yz$$-plane and is a signed distance $$a$$ from it.

Observe that

• the distance from the point $$(x,y,z)$$ to the $$xy$$-plane is $$|z|$$
• the distance from the point $$(x,y,z)$$ to the $$xz$$-plane is $$|y|$$
• the distance from the point $$(x,y,z)$$ to the $$yz$$-plane is $$|x|$$
• the distance from the point $$(x,y,z)$$ to the origin $$(0,0,0)$$ is $$\sqrt{x^2+y^2+z^2}$$

The distance from the point $$(x,y,z)$$ to the point $$(x',y',z')$$ is

$\sqrt{(x-x')^2+(y-y')^2+(z-z')^2} \nonumber$

so that the equation of the sphere centered on $$(1,2,3)$$ with radius $$4\text{,}$$ that is, the set of all points $$(x,y,z)$$ whose distance from $$(1,2,3)$$ is $$4\text{,}$$ is

$(x-1)^2+(y-2)^2+(z-3)^2=16 \nonumber$

This page titled A.15: Cartesian Coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.