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 thatinvolve three-dimensional coordinate geometry will ask you to calculatethe distance between two points in space. There is a general formulathat 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 pointsin coordinate space is basically the same as finding the length of thediagonal of a rectangular solid. In solid geometry, we were given thedimensions of the sides; for coordinate geometry, we have thecoordinates of the endpoints of that diagonal.
Try the example problem below:
|
|
| 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:
