N Dimensional Geometry

Pre-requisites: Dot Product.

The first thing you should know about n dimensional space is that it is absolutely nothing to worry about. You aren't going to be asked to visualize 17 dimensional space or anything freaky like that, because nobody can visualize anything higher than 3 dimensional space (many of us aren't even very good at that). And you can throw out any ideas you might have about the fourth dimension being time or love or what have you, because all it is is an extra number hanging around. To be specific,

Definition: A space is just a set where the elements are called points.

and

Definition: N dimensional space (or Rn for short) is just the space where the points are n-tuplets of real numbers.

You will notice that we are in a sense working backwards: for three dimensional space, we construct cartesian coordinates to get a 3-tuple for every point; now, we forget about the middleman and simply define the point to be the 3-tuple. The origin, in any dimension, is just the n-tuplet (0,0, ... 0).

What about vectors, you ask? Before we defined them to be a magnitude and a direction and then showed how there is a one-to-one correspondence between them and points; now we again invert the order of things and define vectors to be points. Since points are tuples and we know how to add, subtract and scalar multiply tuples, we know how to do all those things for vectors, too.

It is also easy to extend the dot product to vectors in higher dimensions, via the algebraic definition. Just let

( x1, x2, ..., xn ) · ( y1, y2, ..., yn ) = x1 y1 + x2 y2 + ... + xn yn
Having a dot product around allows us to define the length of a vector
|v| = sqrt( v · v )
and the angle between two vectors:
angle = cos-1 ( v &183; w / |v| |w| )

There is no cross product in dimensions greater than 3. For one thing, in dimensions 4 or higher, there are infinitely many unit vectors orthogonal to any given two.

Lines and planes can also be found in higher dimensions, but there isn't often much reason to use them. Before, lines in two or three dimensions could be expressed as

l(t) = OP + t v for P a point and v a vector on the line; the same formula works for higher dimensions. The familiar property of having exactly one line run through two distinct points is maintained.

Planes in three dimensions live a double life: they are both two dimensional flat surfaces and n-1 dimensional flat things. If you want a two dimensional flat surface in n dimension, you are best off using the parametric formula

S(r,s) = OP + r v + s w
If, on the other hand, you want a n-1 dimensional flat thing, you are better off using the implicit formula
A1 x1 + A2 x2 + ... + An xn = B
These are usually called hyperplanes and are useful for approximating the graphs of functions. For example, functions from R to R have graphs in R2 which we approximate using 2 dimensional hyperplanes (i.e., lines).

Exercises:

  1. What is the distance between the points (1,2,3,4) and (-5,2,0,12)?
  2. What is the angle between the vectors (1,0,2,0) and (-3,1,4,-5) ?
  3. Project the point (1,2,3,4,5) onto the vector (7,7,7,1,9).
  4. Give a formula for the line going through (1,2,3,4,5,6) and (0,0,0,17,0,0).
  5. Find the plane containing the three points (1,2,3,4), (2,7,2,7), and (-7,-5,-1,0).
  6. The unit hypercube in four dimensions is described by the equations 0 <= xi <= 1, for i = 1,2,3,4. Give equations for the eight hyperplanes containing the sides of the four dimensional hypercube.
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last modified 1 July 1997