Introduction to 3D Coordinate Geometry

Coordinate geometry, as studied in two dimensions, describes every point of a plane by an ordered pair $(x,y)$ measured against two mutually perpendicular reference lines, the $x$-axis and the $y$-axis, meeting at a fixed origin $O$. This two-dimensional framework is powerful for describing curves that lie flat in a plane -- circles, parabolas, straight lines -- but it cannot describe the position of a point that is not confined to any single plane, such as the tip of an aircraft's wing in flight, a satellite in orbit, or a corner of a room. Describing such points requires one further independent direction of measurement, giving three-dimensional (3D) coordinate geometry -- also called solid geometry or coordinate geometry of space.

Why one more axis is enough. Physical space, as far as classical geometry is concerned, has exactly three independent directions in which a point can be displaced: no combination of movement along any two of them can ever reproduce a displacement along the third. Formally, three mutually perpendicular lines through a common point are enough to locate any point of space uniquely, by measuring how far the point lies from each of the three lines (or, equivalently, from each of the three planes each pair of these lines determines). Extending the 2D idea, three coordinates -- conventionally written $(x, y, z)$ -- are therefore both necessary and sufficient to locate an arbitrary point of space.

What this chapter builds. Starting from three mutually perpendicular coordinate axes meeting at an origin $O$ (Section 2), this chapter develops, in order: how to write down the coordinates of a point in space and measure the straight-line distance between two such points (Section 3); how to describe the direction a line in space points in, using direction cosines and direction ratios (Section 4); how to write down the full equation of a straight line in space, in both vector and Cartesian form, given either a point and a direction or two points on the line (Section 5); the surprising fact that two lines in space can be neither parallel nor intersecting at all -- called skew lines, a phenomenon with no counterpart in plane geometry (Section 6); how to compute the shortest possible distance between two such lines (Section 7); and how to measure the angle between any two lines in space from their direction ratios alone (Section 8).

A first contrast with plane geometry. In a single plane, any two distinct straight lines are related in exactly one of two ways: either they are parallel (never meeting), or they intersect at exactly one point. In three-dimensional space, a third possibility opens up -- two lines can point in genuinely different directions and still never meet, simply because space gives them room to pass by one another without crossing. This is the single biggest conceptual difference between plane and solid coordinate geometry, and it is the reason the notion of skew lines, together with the idea of measuring the shortest distance between two lines that never touch, is central to this chapter.