Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry Chapter 1 Topics in Analytic Geometry
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 62Solution As the parameter t increases, the value of z = ct also increases, so the point(x,y,z) moves upward. However, as t increases, the point (x,y,z) also moves in a pathdirectly over the circlex = acosθ, y = asinθin the xy-plane. The combination of these upward and circular motions produces a corkscrew-shaped curve that wraps around a right circular cylinder of radius a centered on thez-axis. This curve is called a circular helix.zxy✠Parametric Equations for Intersections of SurfacesCurve in 3-space often arise as intersections of surfaces. One method for finding parametricequations for the curve of intersection is to choose one of the variables as the parameter anduse the two equations to express the remaining two variables in terms of that parameter.For example, suppose we want to find parametric equations of the intersection of thecylinder z = x 3 and y = x 2 . We choose x = t as the parameter and substitute this into theequations z = x 3 and y = x 2 , then we obtain the parametric equationsThis curve is called a twisted cubic.x = t, y = t 2 , z = t 3 (4.3)
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 63Vector-Valued FunctionsThe twisted cubic defined by the equations in (4.3) is the set of points of the form (t,t 2 ,t 3 )for real values of t. If we view each of these points as a terminal point for a vector r whoseinitial point is at the origin,r = 〈x,y,z〉 = 〈t,t 2 ,t 3 〉 = ti+t 2 j+t 2 kthen we obtain r as a function of the parameter t, that is, r = r(t). Since this functionproduces a vector, we say that r = r(t) defines r as a vector-valued function of a realvariable, or more simply, a vector-valued function.If r(t) is a vector-valued function in 2-space, then for each allowable value of t the vectorr = r(t) can be represented in terms of components asr = r(t) = 〈x(t),y(t)〉 = x(t)i+y(t)jThe functions x(t) and y(t) are called the component functions or the components ofr(t). Similarly, the component functions of a vector-valued functionin 3-space are x(t), y(t) and z(t).r = r(t) = 〈x(t),y(t),z(t)〉 = x(t)i+y(t)j+z(t)kExample 4.3 The component functions ofarer(t) = 〈t,t 2 ,t 3 〉 = ti+t 2 j+t 3 kx(t) = t, y(t) = t 2 , z(t) = t 3The domain of a vector-valued function r(t) is the set of allowable values for t. If r(t)is defined in terms of component functions and the domain is not specified explicitly, thenthe domain is the intersection of the natural domains of the component functions; this iscalled the natural domain of r(t).Example 4.4 Find the natural domain ofSolution .........r(t) = 〈ln|t−1|,e t , √ t〉 = (ln|t−1|)i+e t j+ √ tkGraphs of Vector-Valued FunctionsIf r(t) is a vector-valued function in 2-space or 3-space, then we define the graph of r(t) tobe the parametric curve described by the component functions for r(t).For example, ifr(t) = 〈1−t,3t,2t〉 = (1−t)i+3tj+2tk (4.4)then the graph of r = r(t) is the graph of the parametric equationsx = 1−t, y = 3t, z = 2tThus, the graph of (4.4) is the line in Example 4.1.✠
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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 62Solution As the parameter t <strong>in</strong>creases, the value of z = ct also <strong>in</strong>creases, so the po<strong>in</strong>t(x,y,z) moves upward. However, as t <strong>in</strong>creases, the po<strong>in</strong>t (x,y,z) also moves <strong>in</strong> a pathdirectly over the circlex = acosθ, y = as<strong>in</strong>θ<strong>in</strong> the xy-plane. The comb<strong>in</strong>ation of these upward and circular motions produces a corkscrew-shaped curve that wraps around a right circular cyl<strong>in</strong>der of radius a centered on thez-axis. This curve is called a circular helix.zxy✠Parametric Equations for Intersections of SurfacesCurve <strong>in</strong> 3-space often arise as <strong>in</strong>tersections of surfaces. One method for f<strong>in</strong>d<strong>in</strong>g parametricequations for the curve of <strong>in</strong>tersection is to choose one of the variables as the parameter anduse the two equations to express the rema<strong>in</strong><strong>in</strong>g two variables <strong>in</strong> terms of that parameter.For example, suppose we want to f<strong>in</strong>d parametric equations of the <strong>in</strong>tersection of thecyl<strong>in</strong>der z = x 3 and y = x 2 . We choose x = t as the parameter and substitute this <strong>in</strong>to theequations z = x 3 and y = x 2 , then we obta<strong>in</strong> the parametric equationsThis curve is called a twisted cubic.x = t, y = t 2 , z = t 3 (4.3)
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