Math101 · Independent Ontario learning library
Math101Three-Dimensional Coordinate System
A rigorous, example-driven guide to three-dimensional coordinate system, including hypotheses, method choice, verification, and practice.
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
- Sketch axes with a consistent right-handed orientation.
- Translate verbal constraints into coordinate equations or inequalities.
- Use vector subtraction before distance or direction calculations.
- Complete squares to identify sphere centers and radii.
- Check dimensions and substitute representative points.
