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# asn1-soln - MAT237Y Multivariable Calculus Summer 2009...

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MAT237Y – Multivariable Calculus Summer 2009 Assignment # 1 Solutions. Questions are worth 10 marks each. Due Tuesday, May 19, at 6:10pm sharp. 1. (a) Using the dot product, show that for x, y R n , the formula 2 k x k 2 + 2 k y k 2 = k x + y k 2 + k x - y k 2 holds. (b) The norm on R n can be defined in terms of the dot product by the formula k x k = x x . Show that the reverse is true. That is, find a formula for x y involving the norms of vectors ( k x k , k y k , k x + y k , and k x - y k for example), and without using coordinates. Solution. (a) Consider ( x + y ) ( x + y ). By definition of the norm, this is equal to k x + y k 2 . But, from the properties of the dot product, we know the following: ( x + y ) ( x + y ) = x x + 2( x y ) + y y = k x k 2 + 2( x y ) + k y k 2 . So we have k x + y k 2 = k x k 2 + 2( x y ) + k y k 2 . (1) We can do a similar calculation to ( x - y ) ( x - y ), and we end up with the following equation: k x - y k 2 = k x k 2 - 2( x y ) + k y k 2 . (2) Add Equations 1 and 2 together, and we end up with k x + y k 2 + k x - y k 2 = 2 k x k 2 + 2 k y k 2 which is what we are trying to prove. This is called the Parallel-

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asn1-soln - MAT237Y Multivariable Calculus Summer 2009...

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