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Unformatted text preview: Math 17C
Kouba ThreeDimensional (3D) Space RECALL : Consider two points (x 1, y 1) and (x 2, y 2) in twodimensional
space. The midpoint of the line segment joining these two points is given by (x1+x2:Y1+Y2) (x1,y1)
2 2 0 "\ ThembeMeen (x,,y1) (xz.y2)
these two points is ' ~ . D=¢(X2~X1‘)2+(Y2y1)2. “" (X2»Y2) RECALL : The set of all points (x, y) in twodimensional space which are a
distance r from a fixed point (h, k) is a circle (with center (h, k) and
radius r) given by the equation (xh)2+(yk)2 = r2. Let (x 1, y 1, z 1) and (x 2. y 2, z 2) be two points in threedimensional
space. The midpoint of the line segment joining these two points is given by (XI+XZIY1+y2,Zi+22> (x,,y1,z1)
2 2 2 ’ .‘\\ [
““\
Thesﬂstamebetween (Xvi/1J1) “ (xz.y2.22)
these two points is ’x
o ‘\ [himx,)2+(y2y,)2+(22—z,)2 ‘ DEFINITION : The set of all points (x, y, z) in threedimensional space which are a distance r from a fixed point (h, k, I) is a sphere (with center
(h, k, l) and radius r given by the equation (xh)2+(yk)2+(zl)2 = r2. Example : Find the center and radius of each of the following spheres. 1. 2x2+2y2+222 = 32 center (0, 0. 0) radius4
2. x2+y2+22—4x+6y =17 center (2,—3, O)
radius «93 Example : The diameter of a sphere h
Determine an equation for this sphere. as endpoints (1, 3, 0) and (2, 4, 6) .
(x+1/2)2+(y7/2)2+(z3)2 = 23/2
Example : Find and simplify an equation for all points (x, y, z) in threedimensional space which are equidistant from the point (1 ,2, 3)
and the plane 2 =1. z=1/a(x—1)2+1/8(y+2)2+1 ...
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 Summer '09
 Lewis

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