-
right-hand rule: Curl the fingers of your right hand around the
z-axis with your thumb pointing in the positive z-direction, then open
your hand 90
: your fingers point in the positive x-direction; closed,
they point in the positive y-direction. - coordinate planes: analogous to the coordinate axes in 2-dimensions: the surfaces created when one of the three coordinates is set to zero. Each coordinate plane will contain two of the three coordinate axes.
- octants: space is now divided into 8 sections, by the walls of the coordinate planes.
- three-dimensional coordinate system: Once coordinate axes are established, coordinates (x,y,z) indicate a point in space. There is a one-to-one correspondence between space points and these triples.
- coordinates of a point: an ordered triple of the values (x,y,z) indicating the location of the point in the three-dimensional coordinate system.
-
Distance formula in three-dimensions: is a simple analogue of the
formula in two dimensions. The distance
between the points
and
is given by
-
Equation of a Sphere: From the distance formula, we can deduce the
equation for the sphere centered on C(h,k,l) of radius r. The sphere is all
points equidistance from the center, of distance r, which is indicated by
Drawing axes is important:
Figure 1: Top: two legitimate representations; Bottom: sin and
sacrilege! Don't do it! It's not physically realizable.
This section serves as an introduction to 3-dimensional coordinate systems and the representation of space, rather than the plane. We see that some formulas extend in quite a natural way.