Engineering Calculus Notes 35

Engineering Calculus Notes 35 - 1.2. VECTORS AND THEIR...

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Unformatted text preview: 1.2. VECTORS AND THEIR ARITHMETIC 23 → − v +→ − w century; the first elementary exposition was given by Edwin Bidwell Wilson (1879-1964) in 1901 [54], based on lectures by the American mathematical physicist Josiah Willard Gibbs (1839-1903)3 [17]. By a geometric vector in R3 (or R2 ) we will mean an “arrow” which can be moved to any position, provided its direction and length are maintained.4 We will denote vectors with a letter surmounted by an arrow, → like this: − . We define two operations on vectors. The sum of two vectors v → is formed by moving − so that its “tail” coincides in position with the w → − , then forming the vector − + − whose tail coincides with →→ “head” of v v w → − and whose head coincides with that of − (Figure 1.18). If → that of v w → − w → − v Figure 1.18: Sum of two vectors → instead we place − with its tail at the position previously occupied by the w → − and then move − so that its tail coincides with the head of − , → → tail of v v w →→ − + − , and it is clear that these two configurations form a we form w v parallelogram with diagonal →→→→ − +− =− +− v w w v (Figure 5.18). This is the commutative property of vector addition. A second operation is scaling or multiplication of a vector by a number. We naturally define →→ 1− = − v v →→→ 2− = − + − v v v →→→→ − = − + − + − = 2− + − →→ 3v v v v v v and so on, and then define rational multiples by → → → → − = m − ⇔ n − = m− ; w v w v n 3 I learned much of this from Sandro Caparrini [6, 7, 8]. This narrative differs from the standard one, given by Michael Crowe [10] 4 This mobility is sometimes expressed by saying it is a free vector. ...
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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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