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ch13sec1 - 13.1 Vector Functions and Space Curves We have...

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13.1 Vector Functions and Space Curves We have studied functions, , that represent the position of a particle on a line. This notion can be extended to more than 1 dimension. Two functions are required to describe the position of a particle in two dimensions. In three dimensions, 3 functions are required. ( ) x f t = Consider the following two-dimensional vector function: 2cos ,sin 0 2 t t t π = 2 r . The component of r is and the component of is . Hence, we can also describe the vector function by writing x 2cos( ) t y x t r a sin( ) t nd ( ) 2cos ( ) sin . t y t t = = For each t corresponds to a point in the xy plane. You can graph r by plotting these points for by using the parametric plot function on your calculator. The vector is from the origin to a point on the curve. ( ) , t r 0 t ( ) t π -2 -1 1 2 -1 -0.5 0.5 1 r(t) You can see that this vector function traces out an ellipse. If we think of r as representing the position of a particle then ( ) t (1) 2cos(1),sin(1) = (1) r . can also be thought of as a vector. The vector in the plot is r , with its tail starting at the origin.
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