Chap12_Sec5

# Chap12_Sec5 - VECTORS AND THE GEOMETRY OF SPACE 12.5...

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12.5 Equations of VECTORS AND THE GEOMETRY OF SPACE Lines and Planes In this section, we will learn how to: Define three-dimensional lines and planes using vectors.

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EQUATIONS OF LINES A line L in three-dimensional (3-D) space is determined when we know: s A point P 0 ( x 0 , y 0 , z 0 ) on L s The direction of L Let P ( x , y , z ) be an arbitrary point on L. s Let r 0 and r be the position vectors of P 0 and P. That is , they have representations and : OP uuur OP uuur r = < x , y , z > and r 0 = < x 0 , y 0 , z 0 > s Assume that vector v gives the direction of the line L and v = < a , b , c > . Since a and v are parallel vectors, there is a scalar t such that a = t v = < ta , tb , tc > If a vector v = < a , b , c > is used to describe the direction of a line L , then the numbers a , b , and c are called direction numbers of L . 0
EQUATIONS OF LINES Triangle Law: r = r 0 + a ( vector equation ) or < x , y , z > = < x 0 + ta , y 0 + tb , z 0 + tc > i.e., x = x 0 + at y = y 0 + bt z = z 0 + ct where, t These equations are called

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## This note was uploaded on 02/23/2010 for the course MATH 221 taught by Professor Staff during the Spring '08 term at Tulane.

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Chap12_Sec5 - VECTORS AND THE GEOMETRY OF SPACE 12.5...

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