VectorFunctionsText(1) - Vector-valued Functions John E...

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Vector-valued Functions John E. Gilbert, Heather Van Ligten, and Benni Goetz A vector-valued function r ( t ) = x ( t ) i + y ( t ) j = ( x ( t ) , y ( t ) ) assigns the position vector r ( t ) to each value of t in the domain of r i.e. , r ( t ) is the vector with tail at (0 , 0) and head at the point ( x ( t ) , y ( t )) . The graph of r is the path the head of the vector r ( t ) sweeps out as t varies. For example, to any function y = f ( x ) corresponds the vector-valued function r ( x ) = ( x, f ( x ) ) = x i + f ( x ) j as shown to the right. The graph of r is then exactly the same as the graph of y = f ( x ) . Thus all we did before with explicitly defined functions carries over to vector functions. ( x, f ( x )) y = f ( x ) x i + f ( x ) j But vector functions deal equally well with implicitly defined functions f ( x, y ) = 0 . Example 1: if r ( t ) = ( a cos t, a sin t ) , a > 0 fixed , then x ( t ) = a cos t, y ( t ) = a sin t . So by using the trig identity cos 2 t + sin 2 t = 1 and eliminating the variable t , we get x 2 + y 2 = a 2 . The graph of r is thus the circle to the right. x y r ( t ) r (0)
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Example 2: What vector function r ( t ) would start at (2 , 3) when t = 0 and then trace clockwise the circle of radius 4 centered at (2 , 3) as t increases ? Hint: the equation of this circle is ( x 2) 2 + ( y 3) 2 = 16 . Example 1 shows the many ways we shall utilize vector functions: given r ( t ) = ( x ( t ) , y ( t ) ) , the head of the vector r ( t ) traces out a plane curve in 2 -space; by identifying the head of r ( t ) with the point ( x ( t ) , y ( t )) we get a curve in the plane given parametrically as the parameter t varies. We could even forget about vectors and simply think
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