Three-Dimensional Coordinate System

The central idea

Cartesian three-space uses ordered triples $(x,y,z)$ relative to mutually perpendicular coordinate axes. Distance between $P$ and $Q$ is $\|Q-P\|$, midpoint is $(P+Q)/2$, and a sphere centered at $(a,b,c)$ with radius $R$ satisfies $(x-a)^2+(y-b)^2+(z-c)^2=R^2$. Coordinate planes divide space into eight octants.

Definitions, hypotheses, and notation

The equation $x^2+y^2+z^2+Dx+Ey+Fz=G$ can be completed square by square; a negative resulting radius squared means no real sphere, while zero gives one point. Distance from the origin is the norm $\sqrt{x^2+y^2+z^2}$, so spheres are three-dimensional level sets of the norm.

Regions such as $x^2+y^2\le R^2$, $a\le z\le b$ describe solids through combined inequalities. Boundary equations alone describe surfaces, not filled volumes. This distinction becomes essential when setting double versus triple integral bounds.

Conceptual meaning

An ordered triple is a position, while subtracting two positions produces a displacement vector. Equations describe sets of points: fixing one coordinate gives a plane, two gives a line parallel to an axis, and three gives a point.

A dependable method and decision rule

1. Sketch axes with a consistent right-handed orientation.
2. Translate verbal constraints into coordinate equations or inequalities.
3. Use vector subtraction before distance or direction calculations.
4. Complete squares to identify sphere centers and radii.
5. Check dimensions and substitute representative points.

Fully worked example

Graphical or geometric meaning

Coordinate planes $xy$, $xz$, and $yz$ are the zero sets of $z$, $y$, and $x$. A sphere intersects a plane in a circle when the plane is close enough to the center, a point when tangent, and nothing when too far.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute the center and a known radius point.
• Compute squared distance before taking a square root.
• Verify midpoint coordinates average corresponding components.

Coordinates, points, and vectors play different roles

A point $(x,y,z)$ specifies location; a vector $\langle a,b,c\rangle$ specifies displacement and can be translated without changing it. The distance between points is the norm of their difference, and spheres collect points at a fixed distance from a center. Coordinate planes and octants help interpret signs, while projections remove one coordinate. Before applying a formula, identify whether an ordered triple is being used as a point, a position vector, or a free vector. Dot products measure angles and projections; cross products produce oriented perpendicular vectors. Units should match across components when Euclidean length is used. A quick sketch of axes and projections prevents swapping coordinates in later line, plane, and surface equations.

Practice

1. Find distance from $(0,0,0)$ to $(1,2,2)$.
2. Find midpoint of $(1,0,-1)$ and $(3,4,5)$.
3. What plane is $x=0$?

Connections and next steps

• Vectors in Three Dimensions
• Lines in Three Dimensions
• Planes in Three Dimensions