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Unformatted text preview: Unit V: Orthogonality 1. Dot Product and Length Definition: The dot product of two vectors u , v R n is u v = n X j =1 u j v j . The dot product is symmetric: u v = v u , u , v R n . The dot product depends linearly on u and on v : ( a x + b y ) v = a ( x v ) + b ( y v ) , x , y , v R n , a,b R , u ( a x + b y ) = a ( u x ) + b ( u y ) , x , y , u R n , a,b R . Definition: The length of a vector u R n is defined by  u  = u u = n X j =1 u 2 j ! 1 / 2 , where we take the nonegative square root. Thus  u  2 = u u = n X j =1 u 2 j . Lemma. The length of a vector in R n has the following properties: (1)  u  0, (2)  u  = 0 if and only if u = , (3)  c u  =  c  u  for u 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 R n , then  u v   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  u + t v  2 = ( u + t v ) ( u + t v ) , and we expand the righthand side, to obtain  u  2 + 2 t u 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 v  v  2 . We plug this value of t into the inequality, and we obtain  u  2 2 ( u v ) 2  v  2 + ( u v ) 2  v  4  v  2 =  u  2 ( u v ) 2  v  2 . Thus ( u v ) 2  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 between 0 and such that cos = u v  u  v  . Then u v =  u  v  cos . Note that = 0 if and only if both u 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, = if and only if u and v are negative multiples of each other. Theorem 2 (Triangle Inequality). For any two vectors u , v R n ,  u + v   u  +  v  . Proof. Use  u + v  2 = ( u + v ) ( u + v ) =  u  2 + 2 u v +  v  2 and the CauchySchwarz inequality, to obtain  u + v  2  u  2 + 2  u  v  +  v  2 = (  u  +  v  ) 2 ....
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 Spring '08
 lee
 Vectors, Dot Product

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