Engineering Calculus Notes 28

Engineering Calculus Notes 28 - on R 2 , in which the...

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16 CHAPTER 1. COORDINATES AND VECTORS and divides AB into two intervals, with respective lengths | AD | = x | DB | = y so x + y = c. Finally, set | CD | = z. Now show the following: x = a sin α y = b sin β z = a cos α = b cos β and use this, together with Pythagoras’ Theorem, to conclude that a 2 + b 2 = x 2 + y 2 + 2 z 2 c 2 = x 2 + y 2 + 2 xy and hence c 2 = a 2 + b 2 2 ab cos( α + β ) . See Exercise 16 for the version of this which appears in Euclid. 14. Oblique Coordinates: Consider an oblique coordinate system
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Unformatted text preview: on R 2 , in which the vertical axis is replaced by an axis making an angle of radians with the horizontal one; denote the corresponding coordinates by ( u,v ) (see Figure 1.11 ). (a) Show that the oblique coordinates ( u,v ) and rectangular coordinates ( x,y ) of a point are related by x = u + v cos y = v sin ....
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