Engineering Calculus Notes 97

Engineering Calculus Notes 97 - 85 1.6. CROSS PRODUCTS...

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Unformatted text preview: 85 1.6. CROSS PRODUCTS triangle) is the projection of the oriented area A (△ABC ) (as a vector) onto the direction normal to P ′ . Note in particular that when △ABC is parallel to P ′ , its oriented area is unchanged, while if △ABC is perpendicular to P ′ , its projection is a degenerate triangle with zero area. As an example, let us consider the projections onto the coordinate planes of the triangle with vertices A(2, −3, 4), B (4, −2, 5), and C (3, −1, 3), which is the triangle we sketched in Figure 1.41. We reproduce this in Figure 1.46, showing the projections of △ABC on each of the coordinate axes pr o jy z (△ AB A C) z B C jxz BC ) p ro j x y( △ A B C ) o pr A (△ y x Figure 1.46: Projections of △ABC The projection onto the xy -plane has vertices A(2, −3), B (4, −2), and C (3, −1), which is the triangle we sketched in Figure 1.36. This has signed area 3/2, so its oriented area is → 3− k 2 —that is, the area is 3/2 and the orientation is counterclockwise when seen from above the xy -plane. The projection onto the yz -plane has vertices A(−3, 4), B (−2, 5), and C (−1, 3) (Figure 1.37) and we saw that its signed area is −1/2. If we look ...
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