c1s2 - 1 1.2 The Norm and Dot Product Chapter 1 Vectors...

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1.2 The Norm and Dot Product 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.2. The Norm and Dot Product Definition 1.5. Let v = [ v 1 , v 2 , . . . , v n ] R n . The norm or magnitude of v is v = v 2 1 + v 2 2 + · · · + v 2 n = n l =1 ( v l ) 2 . Theorem 1.2. Properties of the Norm in R n . For all v, w R n and for all scalars r R , we have: 1. v 0 and v = 0 if and only if v = 0 . 2. rv = | r | v . 3. v + w v + w (the Triangle Inequality). Note. 1 and 2 are easy to see and we will prove 3 later in this section.
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1.2 The Norm and Dot Product 2 Note. A picture for the Triangle Inequality is: 1.2.22, page 22 Definition. A vector with norm 1 is called a unit vector . When writing, unit vectors are frequently denoted with a “hat”: ˆ i. Example. Page 31 number 8. Definition 1.6. The dot product for v = [ v 1 , v 2 , . . . , v n ] and w = [ w 1 , w 2 , . . . , w n ] is v · w = v 1 w 1 + v 2 w 2 + · · · + v n w n = n l =1 v l w l .
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1.2 The Norm and Dot Product 3 Notice. If we let θ be the angle between nonzero vectors v and w , then we get by the Law of Cosines: 1.2.24, page 23 v 2 + w 2 = v w + 2 v w cos θ or ( v 2 1 + v
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