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Engineering Calculus Notes 95

# Engineering Calculus Notes 95 - △ A ′ B ′ C ′ with...

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1.6. CROSS PRODUCTS 83 The (orthogonal) projection of points in R 3 to a plane P takes a point P R 3 to the intersection with P of the line through P perpendicular to P (Figure 1.43 ). We denote this by P proj P P P Figure 1.43: Projection of a Point P on the Plane P P = proj P P. Similarly, a vector −→ v is projected onto the direction of the line where the plane containing −→ v and the normal to P meets P (Figure 1.44 ). −→ v proj P −→ v P Figure 1.44: Projection of a Vector −→ v on the Plane P Suppose ABC is an oriented triangle in R 3 ; its projection to P is the oriented triangle
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Unformatted text preview: △ A ′ B ′ C ′ , with vertices A ′ = proj P ′ A , B ′ = proj P ′ B , and C ′ = proj P ′ C . What is the relation between the oriented areas of these two triangles? Let P be the plane containing △ ABC and let −→ n be the unit vector (normal to P ) such that V A ( △ ABC ) = A −→ n where A is the area of △ ABC . If the two planes P and P ′ are parallel, then △ A ′ B ′ C ′ is a parallel translate of △ ABC , and the two oriented areas...
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