The Three Dimensional Coordinate System
I. Go Over Syllabus
II. xyz-space
In two dimensions, we work in the xy-plane. Analogously, for three dimensions, an xyz space is defined with three axes and coordinate planes (the xy-plane, the xz-plane, and the yz-plane)
III. The Distance Formula and the Equation of a Sphere
There is an extension of the distance formula:
Example: Find the distance from the point (2,1,3) to the point (-1,3,-2)
Solution: D = sqrt[9 + 4 + 25] = sqrt[38]
Recall that a circle is defined as the collection of points in the plane a fixed distance from a central point. Similarly a sphere is defined as the collection of points in space a fixed distance from a central point. In terms of the distance formula we have
r = sqrt[(x - x1)2 + (y - y1)2 + (z - z1)2] or
(x - x1)2 + (y - y1)2 + (z - z1)2 = r2
Exercise: Find the center and radius of the sphere x2 + y2 + z2 - 5x = 0
IV. Traces of Surfaces
The trace of a surface is defined as the intersection of the surface with the coordinate plane.
Example Sketch the xz-trace of the sphere (x + 2)2 + (y + 3)2 + (z - 4)2 = 25
Note that this is a sphere of radius 5 centered at (-2,-3,4) to find the xz-trace, set y = 0 to get
(x + 2)2 + (0 + 3)2 + (z - 4)2 = 25 or (x + 2)2 + (z - 4)2 = 16 which is a circle of radius 4 centered at (-2,4)