Engineering Calculus Notes 96

Engineering Calculus Notes 96 - ℓ have the same length...

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84 CHAPTER 1. COORDINATES AND VECTORS are the same. Suppose the two planes are not parallel, but meet at (acute) angle θ along a line (Figure 1.45 ). P P A B C A B C vector A ( ABC ) −→ n 1 −→ n 2 θ θ Figure 1.45: Projection of a Triangle Then a vector −→ v bardbl parallel to (and hence to both P and P ) is unchanged by projection, while a vector −→ v parallel to P but perpendicular to projects to a vector proj P −→ v parallel to P , also perpendicular to , with length | proj P −→ v | = | −→ v | cos θ. The angle between these vectors is the same as between −→ n and a unit vector −→ n normal to P ; the oriented triangle A B C is traversed counterclockwise when viewed from the side of P determined by −→ n . Furthermore, if ABC has one side parallel to and another perpendicular to , then the same is true of A B C ; the sides parallel to
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Unformatted text preview: ℓ have the same length, while projection scales the side perpendicular to ℓ —and hence the area—by a factor of cos θ . Since every triangle in P can be subdivided (using lines through the vertices parallel and perpendicular to ℓ ) into triangles of this type, the area of any triangle △ ABC is multiplied by cos θ under projection. This means V A ( △ A ′ B ′ C ′ ) = ( A cos θ ) −→ n ′ which is easily seen to be the projection of V A ( △ ABC ) onto the direction normal to the plane P ′ . We have shown Proposition 1.6.4. For any oriented triangle △ ABC and any plane P ′ in R 3 , the oriented area of the projection △ A ′ B ′ C ′ of △ ABC onto P ′ (as a...
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