Formula sheet: Three Dimensional Geometry

1.1 Coordinate axes and planes

Three mutually perpendicular lines through a common point O form the coordinate axes: the x-axis, the y-axis and the z-axis. The point O is the origin. Any two axes determine a plane, so the xy-plane, the yz-plane and the zx-plane are the three coordinate planes. These planes divide all of space into eight regions called octants, numbered with Roman numerals from I to VIII.

A point in space is written as an ordered triple (x, y, z). The first coordinate is the distance along the x-axis, the second along the y-axis and the third along the z-axis, each measured with the sign that the octant requires. Finding a point means moving from the origin to the correct position on the x-axis, then parallel to the y-axis, then parallel to the z-axis; that is the three-dimensional extension of plotting in the plane.

1.2 Octant signs

The sign of each coordinate tells the octant. In the first octant all three coordinates are positive. Moving across a coordinate plane changes the sign of one coordinate: crossing the yz-plane flips the sign of x, crossing the zx-plane flips y and crossing the xy-plane flips z. Two sign flips put the point in an octant that shares only one sign with the first.

The xy-plane consists of points with z equal to zero, the yz-plane has x equal to zero and the zx-plane has y equal to zero. On the x-axis both y and z are zero, and similarly for the other two axes.

1.3 Distance and centroid

The distance between two points in space is the square root of the sum of the squared differences of the three coordinates. This is the space version of the Pythagoras theorem: apply the two-dimensional formula in the xy-plane and then combine the result with the z-difference in a right triangle. The distance from the origin to (x, y, z) is simply the square root of x squared plus y squared plus z squared.

The mid-point of the segment joining two points has each coordinate equal to the average of the corresponding coordinates. The centroid of a triangle has each coordinate equal to the average of the three vertices. These formulas extend the plane formulas by adding the z-coordinate in exactly the same way.

Three points are collinear when the distances between them add up properly: the largest of the three pairwise distances equals the sum of the other two. This test is often quicker than computing slopes or ratios in space, and it is the method the exemplar uses in its collinearity questions.