Calculus Notes 13.3 - Calculus-Stewart Dr. Berg Summer 09...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 13.3 13.3 The Dot Product Another useful operation between two vectors is the dot product. This gives the vector space a Euclidean geometry. Definition The dot product , scalar product , or inner product of a = ( a 1 , a 2 , a 3 ) and b = ( b 1 , b 2 , b 3 ) is a b = a 1 b 1 + a 2 b 2 + a 3 b 3 . Note: The dot product in R 2 or R n is analogous. Example A (1,2,3) (4, 1,0) = 4 2 + 0 = 2 Proposition 1) a a = a 2 2) a b = b a 3) a 0 = 0 4) α a β b = αβ ( a b ) 5) a ( b + c ) = a b + a c Note: In a more general vector space, these properties are used to define an inner product. A vector space with an inner product is known as a Hilbert space. Example B Let a = (1,0,2) , b = ( 2,3,1) , and c = (3,1,0) . Then a b = 2 + 0 + 2 = 0 , a c = 3 + 0 + 0 = 3 , and b c = 6 + 3 + 0 = 3 so a (2 b + c ) = 2 a b + a c = 2(0) + 3 = 3 . Geometric Interpretation The Law of Cosines Let θ be the angle opposite a side of a triangle of length c , and let the other sides be of lengths a and b . Then c 2 = a 2 + b 2 2 ab cos . a b c
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Calculus- Stewart Dr. Berg Summer ‘09 Page 2 13.3 For vectors a and b with θ being the angle between them, the Law of Cosines becomes a b 2 = a 2 + b 2 2 a b cos .
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This note was uploaded on 06/06/2010 for the course M 408 taught by Professor Hodges during the Spring '08 term at University of Texas at Austin.

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Calculus Notes 13.3 - Calculus-Stewart Dr. Berg Summer 09...

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