chapter6 - Chapter 6. Three Dimensional Space 6.1 The...

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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 z-axes. 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 z-axes respectively. 2 MA1505 Chapter 6. Three Dimensional Space This is the Cartesian coordinate system for three dimensional space. We also call this space the xyz-space. By convention, we use the right-handed coordi- nate system . A right-handed coordinate system fix the orientation of the axes as follow: If we rotate the x-axis counterclockwise toward the y-axis, then a right-handed screw will move in the positive z direction. 2 3 MA1505 Chapter 6. Three Dimensional Space 6.2 Vectors in xyz-Space A vector is measurable quantity with a magnitude and a direction . It is geometrically represented by an arrow in the xyz-space 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 xyz-space 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 xyz-space 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.

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chapter6 - Chapter 6. Three Dimensional Space 6.1 The...

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