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# 36 - 16.1-16.2 1 The dot product Definition 1.1 A vector...

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16.1-16.2 January 17, 2012 1 The dot product Definition 1.1. A vector space is a set V with two mappings V × V V and R × V V called vector addition and scalar multiplication such that for all a , b V and α , β R i) a + ( b + c ) = ( a + b ) + c ii) a + b = b + a iii) a + 0 = a iv) a + ( - a ) = 0 v) ( αβ ) a = α ( β a ) vi) ( α + β ) a = α a + β b vii) α ( a + b ) = α a + α b viii) 1 a = a Remark 1.1. What about notions of multiplication for vectors? Definition 1.2. An inner product on a vector space V is a function V × V R which associates with each pair ( a , b ) of vectors in V a real number < a , b > which satisfies 1) < a , a >> 0 if a 6 = 0 (positivity) 2) < a , b > = < b , a > (symmetry) 3) < α a + β b , c > = α < a , c > + β < b , c > 1

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Definition 1.3. The dot product which is the usual inner product on the vector space R n is defined by a · b = n i =1 a i b i . Remark 1.2. The dot product yields a notion of length or size of a vector | a | = ( a · a ) 1 / 2 , in R n this notion is the Euclidean norm . Definition 1.4. The Euclidean distance between two vectors a , b R n is defined to be | a - b | .
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