Lesson: Chapter - 8
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-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:
What 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:
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