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Unformatted text preview: Vectors in the Plane John E. Gilbert, Heather Van Ligten, and Benni Goetz Calculus for functions z = f ( x, y ) of two (or more) variables relies heavily on what you already know about the calculus of functions y = f ( y ) of one variable. A few preliminary ideas about vectors, various coordinate systems in two and three variables, as well as more complicated curves in the plane  ones usually defined implicitly  need to be studied too. Its convenient to do this in terms of vector functions . But once weve done that, calculus is the next step! Lets start with vectors  you may have met them already, and youll certainly make good use of them in a number of your other courses! What is a vector: A quantity, be it geometric, scientific or whatever, is a vector so long as it has both a magnitude (or length ) and a direction . For instance, velocity can be described by a vector because it has a magnitude, namely speed , as well as a direction: the wind blows at a speed of 5 mph from the northwest, Joe heads due north at 75 mph in his car, and so on. Displacements provide a different type of example: lets look at where Bob lives in relation to Alice. His house is at point B which is 223 ft., 18 ENE, from Alices house at point A . If we represent this as an arrow from A to B , it determines a displacement vector AB with magnitude the distance from A to B , and direction the direction from A to B . Its natural to represent this vector by an arrow with A the tail and B the head . A B N E 100 ft In general, well usually label vectors by single boldfaced letters like a , v , ... , and so on. Beware: physicists and engineers sometimes use different notation. The length of a vector v will be denoted by  v  ; this length is a positive number except for the zero vector which has length . Of course, not all quantities can be represented as vectors: for instance, mass, temperature and distance have magnitude,...
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 Spring '11
 Gilbert
 Calculus, Vectors

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