01VectorsInPlane

# 01VectorsInPlane - Vectors in the Plane John E Gilbert...

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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. But a few preliminary ideas about vectors and various coordinate systems in two and three dimensions need to be developed before those single variable ideas can be exploited. Once that’s done, then we can get back to calculus! Let’s start with vectors in the plane - 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: 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: 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 . 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 . It’s natural to represent this vector 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-faced letters like a , v , ... , and so on. Beware: physicists and engineers sometimes use diﬀerent notation. The length of a vector v is denoted by v or by | v | ; this length is a positive number except for the zero vector 0 which has length 0 . Of course, not all quantities can be represented as vectors: for instance, mass, temperature and distance have magnitude, but no direction. Such directionless quantities are real numbers

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01VectorsInPlane - Vectors in the Plane John E Gilbert...

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