Engineering Calculus Notes 36

Engineering Calculus Notes 36 - 24 CHAPTER 1. COORDINATES...

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Unformatted text preview: 24 CHAPTER 1. COORDINATES AND VECTORS → − w → − w →+ − v → − v +→ − w → − v → − w → − v Figure 1.19: Parallelogram Rule (Commutativity of Vector Sums) finally, suppose mi →ℓ ni → is a convergent sequence of rationals. For any fixed vector − , if we draw v → − with all their tails at a fixed arrows representing the vectors (mi /ni ) v position, then the heads will form a convergent sequence of points along a → line, whose limit is the position for the head of ℓ− . Alternatively, if we v → − and any positive real number pick a unit of length, then for any vector v → → → r , the vector r − has the same direction as − , and its length is that of − v v v multiplied by r . For this reason, we refer to real numbers (in a vector context) as scalars. If →→→ − =− +− u v w then it is natural to write →→→ − =− −− v u w → w and from this (Figure 1.20) it is natural to define the negative −− of a → − as the vector obtained by interchanging the head and tail of − . → vector w w → − by any negative This allows us to also define multiplication of a vector v real number r = − |r | as → → (− |r |)− := |r | (−− ) v v → —that is, we reverse the direction of − and “scale” by |r |. v Addition of vectors (and of scalars) and multiplication of vectors by scalars have many formal similarities with addition and multiplication of numbers. We list the major ones (the first of which has already been noted above): • Addition of vectors is →→→→ commutative: − + − = − + − , and v w w v ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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