Math 151 Section 1.3 Vector Functions and Parametric Curves Parametric Curve Let x be functions of t, where t is a parameter. As t varies over its domain, the collection of points (x, y) = (x(t), y(t)) defines a parametric curve. Example: Sketch the given parametric curves. A. x t( )= t ! 3y t( )= 2t !1 B. x t( )= 1! 2ty t( )= 2 + 3tfor ! 3 " t < 3 C. x t( )= t +1y t( )= t2! 4Math 151 D. x t( )= ty t( )= 1! t E. x t( )= 2 sin !y t( )= 3cos !Math 151 Vector Functions For each value of t, consider the point (x, y) = (x(t), y(t)) on the parametric curve to be the terminal point of a vector r t( )= x t( ), y t( ) originating from the origin. r defines a vector function. Example: Sketch the given curves as defined by the vector functions. Include the direction of the curve as t increases. A. r t( )= t ! 3,2t !1 B. r t( )= 2 + cost,1+ sin tMath 151 Vector Equation of a Line The vector equation of a line passing through the point r0= x0, y0( ) and parallel to the vector v = v1,v2 is given by r t( )= r0+ tv. The corresponding parametric equations of the line are given by x t( )= x0+ tv1y t( )= y0+ tv2 Example: Find a vector function of the line parallel to the vector 1,4 and passing through the point (−1, 5). Example: Find parametric equations for the line with slope 43 and passing through the point (2, −5).Math 151 Example: An object is moving in the xy-plane and its position after t seconds is given by r t( )= t + 4,t2+ 2. A. What is the position of the object at time t = 2? B. At what time does the object reach the point (7, 11)? C. Does the object pass through the point (9, 20)? D. Eliminate the parameter to find the Cartesian
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