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 \(\bbbr^2\text{.}\) 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 \(\bbbr^3\text{.}\) 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
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
