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Accelerated Integrated Geometry 3.1 Three-Dimensional Space Three-dimensional space or 3-space: formed by the intersection of an x-axis, y-axis, and z-axis. The axes determine three coordinate planes: xy-plane, xz-plane, yz-plane which divide the 3-space into eight octants. Each point in a 3-space is represented by an ordered triple: (x,y,z) * Examples: Use aruler to plot each of the following points on the three-dimensional coordinate system given. Scale: Yn cm. 1. A(1,2,3) 2. B(-3,-1,-6) 3. C(-2,-2,-3) 4. D(4,-1,5) * Steps to drawing a rectangular prism: 1, Plot endpoints of the diagonal 2. Find differences in x, y, and z values: |x, — %| = length of prism l¥e-y,| = width of prism |z, -z,| = height of prism Decide which endpoint is in foreground/background, Decide which endpoint is on right/left. 5. Draw bases and connect height. ald * Examples: Drawing rectangular prisms (er) () vl Badk-right Front Left Seale: Ve cm ye 3 a yz 5. Draw a rectangular prism having a diagonal with endpoints (1,4, -2) and (2,-1,3). * hauger x is front Larger y Ss right x: [a-t}e4 hen ys -t-ole.L wid 234225 Height ree ae z j 6. Draw a rectangular prism having a diagonal with endpoints(3, 2,0) and (0.4.3). Seale:L cm 2:|3-0|=3 Length 2 yeft-g|e3 Width i z:|9-3 \23 Height (0,43) Accelerated Integrated Geometry 3.2 Lines and Planes in 3-Space « Standard Form of a Plane: Ax +By+Cz=D [The graph of a linear equotion in three variables is a plane. In order to sketch a | plane, we must find the intercepts of each axis i « Examples: Find the x-intercept, y-intercept, and z-intercept. Then, graph the plane. 1, -2x+y-3z=6 2, 2x+3y+4z=12 -2x+0-3()=6 -2(0)+y-30)=6 —-2(0)+0-3z=6 ZarB()4H(0)=12 204 Sy+H(O)= 2 “Ree G@ O+y-O=6 0+0-3z2¢ 2nt040212 O+38ytO212 a2 yeG “3n=6 2x212 Sue 12 z=-3 ™ Ig 2 2 3 3 0,6, “3-3 (3,00) (06,0) ze-2 x26 y= 4 - @Ge) (0,4,0) 2atBy t4zs |Z 2(0)430)+ 42212 £ OF+0+42=17 dezi2 44 223 (90,3) « Standard Equation of Sphere: Center at (X,,¥o:Z) ~ (x= x)" +(¥-Yo) +(2- 2) =” ¢ Examples: Write equation of a sphere with the given center and radius. 3. C(1,5,-3) r=5 4. C(4,-2,-6) r=2V7 (ael)PH(y-5¥+ (243)2(s)? G-H}+ (ya2) (246)*=217)*> 9) (249) ( Gx (y-BY+(43)P2 25 (x-4)*+(y42) (2+ 6)*= Yq 
