Engineering Calculus Notes 89

Engineering Calculus Notes 89 - 77 1.6. CROSS PRODUCTS and...

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Unformatted text preview: 77 1.6. CROSS PRODUCTS and its signed area is 1 2 21 1 −1 1 = [(2)(−1) − (1)(1)] 2 1 = [−2 − 1] 2 3 =− ; 2 you can verify from Figure 1.38 that the path A → B → C → A traverses the triangle clockwise. 6 B (4, 5) 5 A(2, 4) 4 3 C (3, 3) 2 1 0 -1 -1 0 1 2 3 4 Figure 1.38: Oriented Triangle △ABC , Negative Orientation These ideas can be extended to polygons in the plane: for example, a quadrilateral with vertices A, B , C and D is positively (resp. negatively) oriented if the vertices in this order are consecutive in the counterclockwise (resp. clockwise) direction (Figure 1.39) and we can define its signed area as the area (resp. minus the area). By cutting the quadrilateral into two triangles with a diagonal, and using Equation (1.26) on each, we can calculate its signed area from the coordinates of its vertices. This will be explored in Exercises 9-13. For the moment, though, we consider a very special case. Suppose we have ...
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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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