Engineering Calculus Notes 84

Engineering Calculus Notes 84 - 72 CHAPTER 1. COORDINATES...

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Unformatted text preview: 72 CHAPTER 1. COORDINATES AND VECTORS A famous formula for the area of a triangle in terms of the lengths of its sides is the second of two area formulas proved by Heron of Alexandria (ca. 75 AD) in his Metrica : A= s(s − a)(s − b)(s − c) (1.25) where a, b and c are the lengths of the sides of the triangle, and s is the semiperimeter 1 s = (a + b + c). 2 Equation (1.27) is known as Heron’s formula, although it now seems clear from Arabic commentaries that it was already known to Archimedes of Syracuse (ca. 287-212 BC). In Exercise 15 and Exercise 16 we will consider both of the area formulas given in the Metrica ; also, in Exercise 5 we will derive a vector formula for the area of a triangle based on the discussion of the distance from a point to a line on p. 52. Here, however, we will concentrate on finding a formula for the area of a triangle in R2 in terms of the coordinates of its vertices. Suppose the vertices are A(a1 , a2 ), B (b1 , b2 ), and C (c1 , c2 ). Using the side AB as the base, we have − − → b = AB and, letting θ be the angle at vertex A, − → h = AC sin θ, so 1− −− →→ AB AC sin θ. 2 To express this in terms of the coordinates of the vertices, note that A (△ABC ) = − − → → → AB = xAB − + yAB − ı where xAB = b1 − a1 yAB = b2 − a2 and similarly − → → → AC = xAC − + yAC − . ı ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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