Contents     [VG.1] [VG.2] [VG.3] [VG.4] [VG.5] [VG.6] [VG.7]

VG.1 Three-Dimensional Coordinate Systems

 

To represent a point on a two-Dimensional plane, two numbers are necessary Hence one needs to use an ordered pair: $(a,b)$. However, to represent a point in three-dimensional space three numbers are necessary. Hence one uses an ordered triple: $(a,b,c)$.


 

Coordinate Axes- Three directed lines through a central point O (the origin) that are perpendicular to each other. These lines are labeled the x-axis, the y-axis, and the z-axis. Usually we think of the x and y axis being horizontal and the z axis being vertical.

 

Coordinate Planes- Three planes determined by the different axes. The xy-plane contains the x- and the y- axes; the yz-plane contains the y- and z- axes, and the xz-plane contains the x- and z- axes. These three planes divide the space into eight parts, called octants.


Coordinates- The three numbers in the ordered triple, $(a,b,c)$. Each number describes the point with reference to a certain axis. For

instance, the number $a$ describes how far the point is along the x-axis. The number $b$ describes how far the point is along the y-axis, and the number $c$ describes how far the point is along the z-axis. This is shown below:


13.1coorsys__7.png




for 3-D animation

 

Projection- A point $P(a,b,c)$ represents a rectangular box. If one drops the point $c$ down perpendicular so that it is on the xy-plane, we get point $Q~(a,b,0)$. This point is a projection of point A onto the xy-plane. To get projections on to different planes, simply change the one of the numbers to 0. So a projection onto the yz-plane would take the form $Q~(0,b,c)$, and a projection onto the xz- plane would take the form $Q~(a,0,c)$.


13.1coorsys__13.png

for 3-D animation

 

Distance Formula in Three-Dimensions

The distance MATH between the points MATH and MATH is:


MATH


13.1coorsys__19.png

for 3-D animation

 

Equation of a Sphere

An equation of a sphere with center $C(h,k,l)$ and radius $r$ is


MATH

In particular, if the center is the origin O, then an equation of the sphere is


MATH





13.1coorsys__26.png

Contents     [VG.1] [VG.2] [VG.3] [VG.4] [VG.5] [VG.6] [VG.7]