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Unformatted text preview: Unit V: Orthogonality 1. Dot Product and Length Definition: The dot product of two vectors u , v âˆˆ R n is u Â· v = n X j =1 u j v j . The dot product is symmetric: u Â· v = v Â· u , u , v âˆˆ R n . The dot product depends linearly on u and on v : ( a x + b y ) Â· v = a ( x Â· v ) + b ( y Â· v ) , x , y , v âˆˆ R n , a,b âˆˆ R , u Â· ( a x + b y ) = a ( u Â· x ) + b ( u Â· y ) , x , y , u âˆˆ R n , a,b âˆˆ R . Definition: The length of a vector u âˆˆ R n is defined by  u  = âˆš u Â· u = n X j =1 u 2 j ! 1 / 2 , where we take the nonegative square root. Thus  u  2 = u Â· u = n X j =1 u 2 j . Lemma. The length of a vector in R n has the following properties: (1)  u  â‰¥ 0, (2)  u  = 0 if and only if u = , (3)  c u  =  c  u  for u âˆˆ R n and c scalar. Definition: A unit vector is a vector u that has unit length,  u  = 1. If v 6 = , then u = v /  v  is a unit vector. Theorem 1 (CauchySchwarz Inequality). If u , v âˆˆ R n , then  u Â· v  â‰¤  u  v  . Equality holds if and only if one of the vectors is a multiple of the other. Proof. We may assume that v 6 = . We start with the inequality â‰¤  u + t v  2 = ( u + t v ) Â· ( u + t v ) , and we expand the righthand side, to obtain â‰¤  u  2 + 2 t u Â· v + t 2  v  2 . 1 For fixed u and v 6 = , this is a quadratic polynomial in t . It attains its minimum at t = u Â· v  v  2 . We plug this value of t into the inequality, and we obtain â‰¤  u  2 2 ( u Â· v ) 2  v  2 + ( u Â· v ) 2  v  4  v  2 =  u  2 ( u Â· v ) 2  v  2 . Thus ( u Â· v ) 2 â‰¤  u  2  v  2 , and this is equivalent to the CauchySchwarz inequality. Equality holds in the CauchySchwarz inequality if and only if the minimum of the function  u + t v  2 is 0. This occurs if and only if u + t v = for some t , and this occurs if and only if u is a multiple of v . Suppose that u and v are nonzero vectors in R n . The CauchySchwarz inequality allows us to define the angle between u and v to be the value of Î¸ between 0 and Ï€ such that cos Î¸ = u Â· v  u  v  . Then u Â· v =  u  v  cos Î¸. Note that Î¸ = 0 if and only if both u Â· v > 0 and equality holds in the CauchySchwarz inequality. This occurs if and only if u and v are positive multiples of each other. Similarly, Î¸ = Ï€ if and only if u and v are negative multiples of each other. Theorem 2 (Triangle Inequality). For any two vectors u , v âˆˆ R n ,  u + v  â‰¤  u  +  v  . Proof. Use  u + v  2 = ( u + v ) Â· ( u + v ) =  u  2 + 2 u Â· v +  v  2 and the CauchySchwarz inequality, to obtain  u + v  2 â‰¤  u  2 + 2  u  v  +  v  2 = (  u  +  v  ) 2 ....
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This note was uploaded on 06/25/2008 for the course MATH 33a taught by Professor Lee during the Spring '08 term at UCLA.
 Spring '08
 lee
 Vectors, Dot Product

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