# hw1 - Physics 570 Homework No 1 1 due Wednesday 27 January...

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Physics 570 Homework No. 1 due Wednesday, 27 January, 2010 1. Consider a standard 2-dimensional vector space, with the usual Cartesian basis set, { ˆ x, ˆ y } , and a particular vector ~ B x - y . Now consider a diﬀerent pair of basis vectors, e 1 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. 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.

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hw1 - Physics 570 Homework No 1 1 due Wednesday 27 January...

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