Engineering Calculus Notes 38

Engineering Calculus Notes 38 - the zero vector does not...

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26 CHAPTER 1. COORDINATES AND VECTORS O −→ v P ( x,y,z ) −→ w −→ v + w Q ( x, y, z ) −→ w Figure 1.21: Componentwise addition of vectors Similarly, if r is any scalar and −→ v = ( x,y,z ), then r −→ v = ( rx,ry,rz ) : a scalar multiplies all entries of the vector . This representation points out the presence of an exceptional vector—the zero vector −→ 0 := (0 , 0 , 0) which is the result of either multiplying an arbitrary vector by the scalar zero 0 −→ v = −→ 0 or of subtracting an arbitrary vector from itself −→ v −→ v = −→ 0 . As a point , −→ 0 corresponds to the origin O itself. As an “ arrow ”, its tail and head are at the same position. As a displacement , it corresponds to not moving at all. Note in particular that
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Unformatted text preview: the zero vector does not have a well-deFned direction —a feature which will be important to remember in the future. From a formal, algebraic point of view, the zero vector plays the role for vector addition that is played by the number zero for addition of numbers : it is an additive identity element , which means that adding it to any vector gives back that vector: −→ v + −→ 0 = −→ v = −→ 0 + −→ v . A ±nal feature that is brought out by thinking of vectors in R 3 as triples of numbers is that we can recover the entries of a vector geometrically. Note...
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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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