Engineering Calculus Notes 37

Engineering Calculus Notes 37 - 25 1.2. VECTORS AND THEIR...

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Unformatted text preview: 25 1.2. VECTORS AND THEIR ARITHMETIC → − u → − w → − v →→→ − =− +− u v w → − u → -− w → − v →→→ − =− −− v u w Figure 1.20: Difference of vectors → →→ →→ → associative: − + (− + − ) = (− + − ) + − . u v w u v w • Multiplication of vectors by scalars →→ → → distributes over vector sums: r (− + − ) = r − + r − , and v w w v → − = r − + s− . → → distributes over scalar sums: (r + s) v v v We will explore some of these properties further in Exercise 3. The interpretation of displacements as vectors gives us an alternative way → to represent vectors. We will say that an arrow representing the vector − v is in standard position if its tail is at the origin. Note that in this case the vector is completely determined by the position of its head, giving us a → natural correspondence between vectors − in R3 (or R2 ) and points v − → → 3 (resp. R2 ). − corresponds to P if the arrow − P from the origin to P ∈R v O → → P is a representation of − : that is, − is the vector representing that v v → 3 which moves the origin to P ; we refer to − as the displacement of R v position vector of P . We shall make extensive use of the correspondence between vectors and points, often denoting a point by its position vector → − ∈ R3 . p Furthermore, using rectangular coordinates we can formulate a numerical specification of vectors in which addition and multiplication by scalars is − → −→ very easy to calculate: if − = OP and P has rectangular coordinates v → (x, y, z ), we identify the vector − with the triple of numbers (x, y, z ) and v → − = (x, y, z ). We refer to x, y and z as the components or entries write v − → → −→ of − . Then if − = OQ where Q = (△x, △y, △z ) (that is, v w → − = (△x, △y, △z )), we see from Figure 1.21 that w →→ − + − = (x + △x, y + △y, z + △z ); v w that is, we add vectors componentwise. ...
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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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