Engineering Calculus Notes 114

Engineering Calculus Notes 114 - are mutually...

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102 CHAPTER 1. COORDINATES AND VECTORS Substituting back into either equation, we get z = 6 so we can use (1 , 4 , 6) as a basepoint; a parametrization of is −→ p ( t ) = ( −→ ı + 4 −→ + 6 −→ k ) + t ( −→ ı + 5 −→ + 7 −→ k ) or x = 1 + t y = 4 + 5 t z = 6 + 7 t. If we try this when the two planes are parallel, we have linearly dependent normals, and their cross product is zero (Exercise 6 in § 1.6 ). In this case, the two left sides of the equations describing the planes are proportional: if the right sides have the same proportion, then we really have only one equation (the second is the Frst in disguise) and the two planes are the same, while if the right sides have a diferent proportion, the two equations
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Unformatted text preview: are mutually contradictory—the planes are parallel, and have no intersection. ±or example, the two equations x − 2 y + 3 z = 1 − 2 x + 4 y − 6 z = − 2 are equivalent (the second is the Frst multiplied by − 2) and describe a (single) plane, while x − 2 y + 3 z = 1 − 2 x + 4 y − 6 z = 0 are contradictory, and represent two parallel, nonintersecting planes. Oriented Volumes In common usage, a cylinder is the surface formed from two horizontal discs in space, one directly above the other, and of the same radius, by joining their boundaries with vertical line segments. Mathematicians...
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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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