Cartesian Coordinates

Pre-requisites: N-Tuples.

There are many ways to study and describe geometric objects in three-dimensional space. One method, pioneered by the Greeks and formalized by Euclid, is to study these objects axiomatically: to define points, lines, planes and other such entities by the axioms they satisfy ("every two distinct points define a line", for instance).

Another method invented by Descartes [ Oresme actually thought of it first, but never gets any credit] reduces geometry to algebra through the following procedure:

It would be hard to overstate the importance of Cartesian coordinates in the history (and practice) of mathematics. Previously unsolvable geometric problems could be converted to long, tedious, but doable calculations in algebra. [Contrast with the Greek era when even simple algebraic formulas were converted into cumbersome geometric figures.] The whole field of trigonometry makes the most sense once you've introduced Cartesian axes. And, much insight is gained into a complicated algebraic formula when the formula can be graphed, which after all is nothing more than drawing all the points with cartesian coordinates of the form (xf(x)). Or, as we will do shortly, of the form (xyf(xy)).


  1. Draw a right-handed and a left-handed coordinate axis system. Be sure to label your axes and to indicate the positive and negative directions on each one.

  2. What are the cartesian coordinates of the origin? What can you say about the cartesian coordinates of a point on the x axis? Of a point in the y-z plane?

  3. Draw a coordinate axis system, and then find and label the following points: (0,6,0), (3,0,-1), (1,2,3).

  4. Prove the three-dimensional Pythagorean theorem: If a point P has coordinates (x1x2x3), then its distance from the origin is

    You may assume the normal two-dimensional Pythagoren theorem.

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Last modified November 20, 1998