# 601 - CHAPTER 6 ORTHOGONALITY Example Perpendicularity in...

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1 CHAPTER 6 ORTHOGONALITY Example: Perpendicularity in R 2 : the Pythagorean Theorem algebraic condition for perpendicularity in R 2

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2 Properties: 1. 0 is orthogonal to every vector in R n . 2. For u , v R 3 , u and v are perpendicular if and only if u v = 0. 3. The dot product can also be represented by the matrix product 4. For u R n , v R m , and A R m × n , Proof All results may be proven easily using the definition. Corollaries: 1. For c R , u , v R n , c u v has a clear meaning: ( c u ) v or c ( u v ). 2. For c 1 , c 2 , , c p R , u , v 1 , v 2 , , v p R n ,
3 Example: Proof = 0 if and only if u and v are orthogonal Corollary: Proof = 0 ⇔| | u || = || v || Orthogonal projection of a vector onto a line v : any vector in R 2 u : any nonzero vector on L w : orthogonal projection of v onto L , w = c u z : v w

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4 and Distance from P (tip of v ) to L : Example: L is y = (1/2) x in R 2 and v = [ 4 1 ] T . Let u = [ 2 1 ] T . Then and the distance from the tip of v to L is * Proof The inequality holds when u = 0 . Assume u 0 . Then . 0 ) ( ) ( : Note . 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = = = +
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601 - CHAPTER 6 ORTHOGONALITY Example Perpendicularity in...

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