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Unformatted text preview: Physics 570 Homework No. 1 Solutions: due Wednesday, 27 January, 2010 1. Consider a standard 2dimensional vector space, with the usual Cartesian basis set, { ˆ x, ˆ y } , and a particular vector ~ B ≡ 3ˆ x 2ˆ y . Now consider a different pair of basis vectors, e 1 ≡ 2ˆ x ˆ y , e 2 ≡ ˆ x + ˆ y . Please determine “the components” of the vector ~ B relative to this basis. However, I want to give you two distinct definitions of the word components. The purpose of the components of a vector is to determine uniquely the vector in question. Both of these definitions serve that purpose. Therefore, I ask that you determine both of them, and then the relation between them. [5 pts] a. The first definition, for the components { B i } 2 1 , is given by the following equality: ~ B = B 1 e 1 + B 2 e 2 . b. The second definition, for the components { B j } 2 1 , is given by the following definitions: B 1 ≡ ~ B · e 1 , B 2 ≡ ~ B · e 2 , where the “dot product” used is the usual, Cartesian dot product that we always use. c. The following matrix W relates the matrix presentation of these two sets of components: B L ≡ B 1 B 2 ¶ , B U ≡ B 1 B 2 ¶ ; B U = W B L W = 1 9 2 1 1 5 ¶ . Please verify this statement and then determine the following matrix M ≡ e 1 · e 1 e 1 · e 2 e 2 · e 1 e 2 · e 2 ¶ , and explain the relationship between the two matrices, M and W . ........................................................................................... a. The simplest way to determine the coefficients { B i } is to first resolve the definitions of e j in terms of ˆ x and ˆ y , insert them into the equation defining ~ B , and solve: ˆ x = 1 3...
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This note was uploaded on 03/08/2010 for the course PHYSICS AN 570 taught by Professor Davids.king during the Spring '10 term at University of Nebraska Medical University.
 Spring '10
 DavidS.King

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