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Unformatted text preview: Chapter 6. Three Dimensional Space 6.1 The Coordinate System of the 3D Space For three dimensional space, we first fix a coordinate system by choosing a point called the origin , and three lines, called the coordinate axes, so that each line is perpendicular to the other two. These lines are called the x. y and zaxes. Associated with a point P in three dimensional space is an ordered triple ( a,b,c ) where a , b and c are the projections of P on the x, y and zaxes respectively. 2 MA1505 Chapter 6. Three Dimensional Space This is the Cartesian coordinate system for three dimensional space. We also call this space the xyzspace. By convention, we use the righthanded coordi nate system . A righthanded coordinate system fix the orientation of the axes as follow: If we rotate the xaxis counterclockwise toward the yaxis, then a righthanded screw will move in the positive z direction. 2 3 MA1505 Chapter 6. Three Dimensional Space 6.2 Vectors in xyzSpace A vector is measurable quantity with a magnitude and a direction . It is geometrically represented by an arrow in the xyzspace with an initial point and a terminal point. The direction of the arrow gives the direction of the vector; and the length of the arrow gives the magnitude of the vector. 6.2.1 Terminologies and notations (1) Let P and Q be points in the xyzspace with co ordinates ( x 1 ,y 1 ,z 1 ) and ( x 2 ,y 2 ,z 2 ) respectively. Then the vector→ PQ is algebraically given by→ PQ = x 2 x 1 y 2 y 1 z 2 z 1 . 3 4 MA1505 Chapter 6. Three Dimensional Space The vector→ OP = x 1 y 1 z 1 is called the position vector of P . (2) The zero vector in the xyzspace is O = . (3) The sum of v 1 = x 1 y 1 z 1 and v 2 = x 2 y 2 z 2 is v 1 + v 2 = x 1 + x 2 y 1 + y 2 z 1 + z 2 . [Note that v 1 + O = O + v 1 = v 1 .] (4) The negative of v 1 = x 1 y 1 z 1 is v 1 =  x 1 y 1 z 1 . [Note that v 1 v 1 = v 1 + v 1 = O .] (5) The difference v 1 v 2 is v 1 v 2 = v 1 +( v 2 ) = x 1 y 1 z 1 +  x 2 y 2 z 2 = x 1 x 2 y 1 y 2 z 1 z 2 . 4 5 MA1505 Chapter 6. Three Dimensional Space (6) If c is a real number, the scalar c v 1 of v 1 by c is c v 1 = cx 1 cy 1 cz 1 . If c > 0, then c v 1 is in the same direction as v 1 . If d < 0, then d v 1 is in the opposite direction as v 1 . (7) The magnitude of v 1 = x 1 y 1 z 1 is  v 1  = q x 2 1 + y 2 1 + z 2 1 . [Note that  c v 1  =  c   v 1  for a real number c .] 5 6 MA1505 Chapter 6. Three Dimensional Space 6.2.2 Example Let P 1 , P 2 , Q 1 and Q 2 be the points (3 , 2 , 1), (0 , , 0), (5 , 5 , 4) and (2 , 3 , 5) respectively....
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This note was uploaded on 05/10/2011 for the course MATH 1505 taught by Professor Yap during the Winter '11 term at National University of Singapore.
 Winter '11
 yap
 Math

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