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Engineering Calculus Notes 15

# Engineering Calculus Notes 15 - x 1,y 1 and x 2,y 2 then we...

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1.1. LOCATING POINTS IN SPACE 3 ( , +) ( , ) (+ , +) (+ , ) Figure 1.2: Direction Conventions (1601-1665) and Ren´ e Descartes (1596-1650). By means of such a scheme, a plane curve can be identi±ed with the locus of points whose coordinates satisfy some equation; the study of curves by analysis of the corresponding equations, called analytic geometry , was initiated in the research of these two men. Actually, it is a bit of an anachronism to refer to rectangular coordinates as “Cartesian”, since both Fermat and Descartes often used oblique coordinates , in which the axes make an angle other than a right one. 1 Furthermore, Descartes in particular didn’t really consider the meaning of negative values for the abcissa or ordinate. One particular advantage of a rectangular coordinate system over an oblique one is the calculation of distances. If P and Q are points with respective rectangular coordinates (
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Unformatted text preview: x 1 ,y 1 ) and ( x 2 ,y 2 ), then we can introduce the point R which shares its last coordinate with P and its ±rst with Q —that is, R has coordinates ( x 2 ,y 1 ) (see Figure 1.3 ); then the triangle with vertices P , Q , and R has a right angle at R . Thus, the line segment PQ is the hypotenuse, whose length | PQ | is related to the lengths of the “legs” by Pythagoras’ Theorem | PQ | 2 = | PR | 2 + | RQ | 2 . But the legs are parallel to the axes, so it is easy to see that | PR | = |△ x | = | x 2 − x 1 | | RQ | = |△ y | = | y 2 − y 1 | and the distance from P to Q is related to their coordinates by | PQ | = r △ x 2 + △ y 2 = r ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . (1.1) 1 We shall explore some of the diFerences between rectangular and oblique coordinates in Exercise 14 ....
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