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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 913.
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.
 Fall '08
 ALL
 Calculus

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