22-Vectors in a Plane

22-Vectors in a Plane - Vectors in the Plane John E....

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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 morecomp l icatedcurvesinthep lane-ones usually deFned implicitly - need to be studied too. It’s convenient to do this in terms of vector functions .Bu ton c ew e ’v edon etha t ,ca l cu lu si sth en ex ts t ep ! Let’s start with vectors - you may have met them already, and you’ll certainly make good use of them in a number of your other courses! What is a vector: Aquant ity ,beitgeometr ic ,sc ient iFco rwhatever ,isa vector so long as it has both a magnitude (or length )anda direction .±o rin s tance , velocity can be described by a vector because it has a magnitude, namely speed ,aswe l lasad irect ion :thew indb lowsataspeedo f5mphfromthe north-west, Joe heads due north at 75 mph in his car, and so on. Displacements provide a di²erent type of example: let’s look at where Bob lives in relation to Alice. His house is at point B which is 223 ft., 18 ENE, from Alice’s house at point A .I fw erep re s t this as an arrow from A to B ,itdete rm inesa displacement vector -→ AB with magnitude the distance from A to B ,and direction the direction from A to B t sna tu ra ltor r e s tth i sv e c to r by an arrow with A the tail and B the head . A B N E 100 ft In general, we’ll usually label vectors by single bold-facedlettersl ike a , v ,. . .,andsoon .Bewa : physicists and engineers sometimes use di²erent notation. The length of a vector v will be denoted by | v | ;th islengthisapos it ivenumberexceptfo rthe zero vector 0 which has length 0 .O
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

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22-Vectors in a Plane - Vectors in the Plane John E....

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