1
11.1: Three-Dimensional Coordinates
Recall: The x-y plane is the set of all points (x0 , y0 ), where x0 refers to the horizontal and y0 refers
to the vertical.
Now: 3-dimensional space is
The distance between 2 points in 3-dimensional space is given by
Proof :
The set of points equidistant from a given point is called a
Therefore, we can derive the equation as follows:
1
Examples:
Describe the set of points which satisfy the equation x2 + 2x + y 2 − y + z 2 = 0.
Describe the set of points in 3-dimensional space which satisfy the equation y = mx + b, where m and
b are constants. Sketch the graph below.
Describe the set of all points which are equidistant from the TWO points (1, 2, 3) and (3, 5, 7). Find
the equation of this object.
2
On Beyond Average:
Find an equation of the plane parallel to the x-axis which contains the points (0, 2, 0) and (0, 0, 2).
Determine if the triangle with vertices P (4, 2, 1), Q(3, 3, 1), and R(3, 2, 3) is scalene, isosceles, or equilateral. Is 4P QR a right triangle?
Describe the set of points in 3-dimensional space which satisfy the equation x2 + z 2 = 4.
3