Sunday, June 17, 2012

Coordinates of a Point in Space

We have already plotted a point on the xy-plane by an ordered pair that consists of two real numbers, (x-coordinate , y-coordinate) , which denote signed distances along the x-axis and y-axis, respectively, from the origin (0,0). These axes, which are collectively referred to as the coordinate axes, divided the plane into four quadrants. 

Let us now generalize these concepts to three-dimensional space, or xyz-space. In this space, a point is represented by an ordered triple (x, y, z) that consists of three numbers, an x-coordiante, a y-coordinate, and a z-coordinate. These coordinates indicate the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, with coordinates (0, 0, 0). 

Three-dimensional space contains infinitely many planes, just as two-dimensional space consists of infinitely many lines. Three planes are of particular importance: the xy-plane, which contains the x-axis and y-axis; the yz-plane, which contains the y-axis and z-axis; and the xz-plane, which contains the x-axis and z-axis.

Alternatively, the xy-plane can be described as the set of all points (x, y, z)for which z=0. Similarly, the yz-plane is the set of all points of the form (0, y, z), while the xz-plane is the set of all points of the form (x, 0, z).

These three planes divide xyz-space into eight octants. Within each octant, all x-coordiantes have the same sign, as do all y-coordinates, and all z-coordinates. In particular, the first octant is the octant in which all three coordinates are positive.

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