Engineering Calculus Notes 107

Engineering Calculus Notes 107 - the Metrica is an...

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1.6. CROSS PRODUCTS 95 with −→ v 0 = −→ v n . Show that the (signed) area of the polygon is 1 2 n s i =1 v v v v x i 1 y i 1 x i y i v v v v . 14. Now extend the deFnition of the area swept out by a line to space, by replacing signed area (in the plane) with oriented area in space: that is, given three points D,A,B R 3 , the area swept out by the line DP as P moves from A to B along −−→ AB is deFned to be the oriented area V A ( DAB ). Show that the oriented area of a triangle ABC R 3 in space equals the area swept out by the line DP as P traverses the triangle, for any point D R 3 . ( Hint: Consider the projections on the coordinate planes, and use Exercise 11 .) History notes: 15. Heron’s First Formula: The Frst area formula given by Heron in
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Unformatted text preview: the Metrica is an application of the Law of Cosines, as given in Book II, Propositions 12 and 13 in the Elements . Given △ ABC , we denote the (lengths of the) side opposite each vertex using the corresponding lower case letter (see ±igure 1.50 ). A B C D c b a A B C D c b ← a → ±igure 1.50: Propositions II.12-13: c 2 = a 2 + b 2 ± 2 c · CD (a) Obtuse Case: Suppose the angle at C is obtuse. Extend BC to the foot of the perpendicular from A , at D . Prove Euclid’s Proposition 11.12: c 2 = a 2 + b 2 + 2 c · CD....
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