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:
