Engineering Calculus Notes 104

# Engineering Calculus Notes 104 - segments to be the sum of...

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92 CHAPTER 1. COORDINATES AND VECTORS (b) Show that the above is not true if B is not between A and C . (c) Show that σ ( O ,A,B ) A ( △O AB ) + σ ( O ,B,C ) A ( △O BC ) + σ ( O ,C,A ) A ( △O CA ) = 0 regardless of the order of A , B and C along the line. 10. Show that the oriented area of a triangle can also be calculated as half of the cross product of the vectors obtained by moving along two successive edges: vector A ( ABC ) = 1 2 −−→ AB × −−→ BC ( Hint: You may use Exercise 8 .) Challenge Problems: Given a point D in the plane, and a directed line segment −−→ AB , we can define the area swept out by the line DP as P moves from A to B along −−→ AB to be the signed area of the oriented triangle [ D,A,B ]. We can then extend this definition to the area swept out by DP as P moves along any broken-line path ( i.e. , a path consisting of finitely many directed line
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Unformatted text preview: segments) to be the sum of the areas swept out over each of the segments making up the path. 11. (a) Show that the area swept out by DP as P travels along an oriented triangle equals the signed area of the triangle: that is, show that σ ( ABC ) A ( △ ABC ) = σ ( DAB ) A ( △ DAB )+ σ ( DBC ) A ( △ DBC )+ σ ( DCA ) A ( △ DCA ) . ( Hint: This can be done geometrically. Consider three cases: D lies outside, inside, or on △ ABC . See ±igure 1.47 .) (b) Show that the area swept out by O P as P moves along the line segment from ( x ,y ) to ( x 1 ,y 1 ) is 1 2 v v v v x y x 1 y 1 v v v v ....
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