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Unformatted text preview: UNIT 2 PARAMETRIC EQUATIONS INTRO: We study parametric equations, their relationship to Cartesian equations and their graphs. We will also study points of intersection of two trajectories, and collision points. 1. Parametric Equations In two dimensions, a path (trajectory) in the x- y plane might be written in Cartesian (rect- angular) coordinates, such as the parabolic path y = x 2 . However, exactly the same parabolic path can be written in terms of a parameter, for example ( x = t y = t 2 ,- < t < . That is to say, rather than giving y as a function of x , both x and y can be given in terms of a parameter, say t : x = f ( t ) , y = g ( t ). We wish to study these parametric equations. In particular, we need to be able to switch between parametric equations and Cartesian equations, to recognize the graphs of parametric equations, to determine if different parametric equations have points of intersection, etc. These issues and others will be discussed in Units 1 and 2. In this Unit and the next, parametric equations will always be in two dimensions. Later in the course, it will be necessary to study trajectories in three dimensions, ie, parametric equations of the form x = f ( t ) , y = g ( t ) , z = h ( t ). In both cases, it is sometimes useful to think of the parameter t as representing the time, and the parametric equations as giving the x and y coordinates (or x, y and z coordinates) of a particle following the trajectory. Thus one sees already a difference between Cartesian and parametric equations. A Cartesian equation such as the parabola y = x 2 may give the trajectory of a particle, but nothing more. Parametric equations such as x = t, y = t 2 give the same trajectory. but also specify exactly when the particle is at each point along the trajectory. Thus one can compute its velocity, acceleration, direction of travel, etc, none of which would be known if only the path y = x 2 were given. Note that the parameter in parametric equations may be restricted to an interval, so, for example, the parametric equations x = t, y = t 2 , < t < 2 describe only the section of the parabola y = x 2 between the points (0 , 0) and (2 , 4). We give some examples of parametric equations, explaining shortly how we can tell that the curves are lines, parabolas, etc. Examples : (1) ( x = 3 t y = 6 t < t < (half-line from the origin to infinity) (2) ( x = 2 t y = t 2- < t < (parabola) VECTOR GEOMETRY 20 GREENBERG (3) ( x = 2 t y = t + 3 t + 4 t < (half-parabola) (4) ( x = sin 4 t- 2 y = 2sin 2 t 2 < t < 3 2 (section of a parabola in y ) What makes this somewhat confusing: the following parametric equations give the same curve as (1)....
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