math125-lecture 5.2 part 2

math125-lecture 5.2 part 2 - Let { v 1 , v 2 } be an...

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Unformatted text preview: Let { v 1 , v 2 } be an orthonormal set. If u is in span{ v 1 , v 2 }, then: u = ( u v 1 ) v 1 + ( u v 2 ) v 2 Sec 5.2.p2 Sunday, February 07, 2010 12:03 AM Section5dot2Part2 Page 1 Let { v 1 , v 2 } be an orthonormal set. If u is in span{ v 1 , v 2 }, then: u = ( u v 1 ) v 1 + ( u v 2 ) v 2 a scalar multiple a scalar multiple of v 1 so parallel of v 2 so parallel to v 1 to v 2 5.2.p2.1 Sunday, February 07, 2010 12:03 AM Section5dot2Part2 Page 2 Let { v 1 , v 2 } be an orthonormal set. If u is in span{ v 1 , v 2 }, then: u = ( u v 1 ) v 1 + ( u v 2 ) v 2 a scalar multiple a scalar multiple of v 1 so parallel of v 2 so parallel to v 1 to v 2 ( u v 2 ) v 2 = projection of u onto v 2 u ( u v 1 ) v 1 = projection of u onto v 1 v 2 v 1 5.2.p2.2 Sunday, February 07, 2010 12:03 AM Section5dot2Part2 Page 3 Section5dot2Part2 Page 4 Let { v 1 , v 2 } be an orthonormal set. If u is in span{ v 1 , v 2 }, then: u = ( u v 1 ) v 1 + ( u v 2 ) v 2 a scalar multiple a scalar multiple of v 1 so parallel of v 2 so parallel to v 1 to v 2 If v is a unit vector, then the projection of u onto v is defined to be Proj v ( u ) = ( u v ) v Proj v ( u ) is a new vector parallel to v ( u v 2 ) v 2 = projection of u onto v 2 u ( u v 1 ) v 1 = projection of u onto v 1 v 2 v 1 5.2.p2.3 (*) Sunday, February 07, 2010 12:03 AM Section5dot2Part2 Page 5 What if v is NOT a unit vector? It still makes sense to look for the projection of u onto v , but to use the "theory" from the u previous page we need a unit vector w in the direction of v : v w 5.2.p2.4 Sunday, February 07, 2010 12:13 AM Section5dot2Part2 Page 6 What if v is NOT a unit vector? It still makes sense to look for the projection of u onto v , but to use the "theory" from the u previous page we need a unit vector w in the direction of v : v v We normalize: w = | v | w 5.2.p2.5 Sunday, February 07, 2010 12:13 AM Section5dot2Part2 Page 7 What if v is NOT a unit vector? It still makes sense to look for the projection of u onto v , but to use the "theory" from the u previous page we need a unit vector w in the direction of v : v v We normalize: w = | v | w v v Then Proj v ( u ) = Proj w ( u ) = ( u w ) w = ( u ) | v | | v | u v = ( ) v | v | 2 5.2.p2.6 Sunday, February 07, 2010 12:13 AM Section5dot2Part2 Page 8 What if v is NOT a unit vector?...
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math125-lecture 5.2 part 2 - Let { v 1 , v 2 } be an...

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