Coordinate Space
When we add another dimension to the coordinate plane,
creating a coordinate space, a new axis must be introduced. Meet
the z-axis:
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-dimensionsal
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, 1)
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 as measurements; for coordinate geometry we have the
coordinates of the endpoints of that diagonal.
Try the example problem below:
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What
is the distance between the points (4, 1, –5) and (–3, 3, 6)? |
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Using the formula, the answer is
, which approximately equals 13.19. To
see this in diagram form, take a look at the figure below:
