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Unformatted text preview: 6.1 Projections 1 Chapter 6. Orthogonality 6.1 Projections Note. We want to find the projection ~p of vector ~ F on sp( ~a ): Figure 6.1, Page 327. We see that ~p is a multiple of ~a . Now (1 / k ~a k ) ~a is a unit vector having the same direction as ~a , so ~p is a scalar multiple of this unit vector. We need only find the appropriate scalar. From the above figure, we see that the appropriate scalar is k ~ F k cos , because it is the length of the leg labeled ~p of the right triangle. If ~p is in the opposite direction of ~a and [ / 2 , 2 ]: 6.1 Projections 2 Figure 6.2, Page 327. then the appropriate scalar is again given by k ~ F k cos . Thus ~p = k ~ F k cos k ~a k ~a = k ~ F k ~a k cos k ~a k ~a k ~a = ~ F ~a ~a ~a ~a. We use this to motivate the following definition. Definition. Let ~a, ~ b R n The projection ~p of ~ b on sp ( ~a ) is ~p = ~ b ~a ~a ~a ~a....
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- Fall '04