Engineering Calculus Notes 78

Engineering Calculus Notes 78 - vector-valued function...

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66 CHAPTER 1. COORDINATES AND VECTORS Suppose now we want to describe a plane P parallel to P 0 ( −→ v , −→ w ), but going through an arbitrarily given basepoint P 0 ( x 0 ,y 0 ,z 0 ). If we let −→ p 0 = x 0 −→ ı + y 0 −→ + z 0 −→ k be the position vector of P 0 , then displacement by −→ p 0 moves the origin O to P 0 and the plane P 0 ( −→ v , −→ w ) to the plane P through P 0 parallel to P 0 ( −→ v , −→ w ). It is clear from Remark 1.5.3 that the position vector −→ p = x −→ ı + y −→ + z −→ k of every point in P can be expressed as −→ p 0 plus some linear combination of −→ v and −→ w −→ p ( s,t ) = −→ p 0 + s −→ v + t −→ w or x = x 0 + sv 1 + tw 1 y = y 0 + sv 2 + tw 2 z = z 0 + sv 3 + tw 3 for a unique pair of scalars s,t R . These scalars form an oblique coordinate system for points in the plane P . Equivalently, we can regard these equations as deFning a
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Unformatted text preview: vector-valued function −→ p ( s,t ) which assigns to each point ( s,t ) in the “ st-plane” a point −→ p ( s,t ) of the plane P in R 3 . This is a parametrization of the plane P ; by contrast with the parametrization of a line, which uses one parameter t , this uses two parameters, s and t . We can use this to parametrize the plane determined by any three noncollinear points. Suppose △ PQR is a nondegenerate triangle in R 3 . Set −→ p = −−→ O P, the position vector of the vertex P , and let −→ v = −−→ PQ and −→ w = −→ PR...
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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.

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