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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 north-west, 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 bold-faced 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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