When we add another dimension to the coordinate plane,
creating a coordinate space, a new axis must be introduced. Meet
The z-axis is perpendicular to both the x-
and y-axes. A point in three dimensions is specified
by three coordinates: (x, y, z).
The only questions you’re likely to see that involve three-dimensional
coordinate geometry will ask you to calculate the distance between
two points in space. There is a general formula that allows you
to make such a calculation. If the two points are (x1, y1, z1)
and (x2, y2, z2),
then the distance between them is:
Determining the distance between two points in coordinate
space is basically the same as finding the length of the diagonal
of a rectangular solid. In solid geometry, we were given the dimensions
of the sides; for coordinate geometry, we have the coordinates of
the endpoints of that diagonal.
Try the example problem below:
is the distance between the points (4, 1, –5) and (–3, 3, 6)?
Using the formula, the answer is
, which approximately equals 13.19. To
see this in diagram form, take a look at the figure below: