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# c6s1 - 6.1 Projections 1 Chapter 6 Orthogonality 6.1...

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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 / a ) 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 F cos θ , because it is the length of the leg labeled p of the right triangle. If p is in the opposite direction of a and θ [ π/ 2 , 2 π ]:

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6.1 Projections 2 Figure 6.2, Page 327. then the appropriate scalar is again given by F cos θ . Thus p = F cos θ a a = F a cos θ a a 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. Example. Page 336 number 4.
6.1 Projections 3 Definition 6.1. Let W be a subspace of R n . The set of all vectors in R n that are orthogonal to every vector in W is the orthogonal complement of W and is denoted by W .

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c6s1 - 6.1 Projections 1 Chapter 6 Orthogonality 6.1...

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