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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
o jy z (△ AB A C) z B C
jxz BC ) p
ro j x y( △ A B C ) o
(△ 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
—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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- Fall '08