13.1 Vector Functions and Space Curves

1 Copyright Cengage Learning. All rights Functions andSpace Curves22 Vector Functions and Space CurvesIn general, a function is a rule that assigns to each elementin the domain an element in the Vector -valued function, or Vector function, is simply afunction whose domain is a set of real numbers and whoserange is a set of are most interested in Vector Functions r whose valuesare three-dimensional means that for every number t in the domain of r thereis a unique Vector in V3 denoted by r(t).33 Vector Functions and Space CurvesIf f (t), g(t), and h(t) are the components of the Vector r(t),then f, g, and h are real-valued Functions called thecomponent Functions of r and we can write r(t) = f (t), g(t), h(t) = f (t) i + g(t) j + h(t) kWe use the letter t to denote the independent variablebecause it represents time in most applications of 1If r(t) = t 3, ln(3 t), then the component Functions are f (t) = t 3 g(t) = ln(3 t) h(t) =By our usual convention, the domain of r consists of allvalues of t for which the expression for r(t) is expressions t 3, ln(3 t), and are all defined when3 t > 0 and t the domain of r is the interval [0, 3).]

2 55 Limits and ContinuityThe limit of a Vector function r is defined by taking thelimits of its component Functions as of Vector Functions obey the same rules as limits ofreal-valued and ContinuityA Vector function r is continuous at a ifIn view of Definition 1, we see that r is continuous at a ifand only if its component Functions f, g, and h arecontinuous at CurvesThere is a close connection between continuous vectorfunctions and Space that f, g, and h are continuous real-valuedfunctions on an interval the set C of all points (x, y, z) in Space , where x = f (t) y = g(t) z = h(t)and t varies throughout the interval I, is called a CurvesThe equations in (2) are called parametric equations of Cand t is called a can think of C as being traced out by a moving particlewhose position at time t is (f (t), g(t), h(t)).If we now consider the Vector function r(t) = f (t), g(t), h(t) ,then r(t) is the position Vector of the point P(f (t), g(t), h(t))on CurvesThus any continuous Vector function r defines a spacecurve C that is traced out by the tip of the moving vectorr(t), as shown in Figure 1C is traced out by the tip of a movingposition Vector r(t).

3 1010 Example 4 Sketch the curve whose Vector equation isr(t) = cos t i + sin t j + t kSolution:The parametric equations for this curve are x = cos t y = sin t z = tSince x2 + y2 = cos2t + sin2t = 1, the curve must lie on thecircular cylinder x2 + y2 = point (x, y, z) lies directly above the point (x, y, 0),which moves counterclockwise around the circle x2 + y2 = 1in the 4 Solution(The projection of the curve onto the xy-plane has vectorequation r(t) = cos t, sin t, 0 .) Since z = t, the curvespirals upward around the cylinder as t increases. Thecurve, shown in Figure 2, is called a dFigure 21212 Space CurvesThe corkscrew shape of the helix in Example 4 is familiarfrom its occurrence in coiled also occurs in the model of DNA (deoxyribonucleic acid,the genetic material of living cells).In 1953 James Watson andFrancis Crick showed thatthe structure of the DNAmolecule is that of two linked,parallel helixes that areintertwined as in Figure double helixFigure 31313 Using Computers to Draw Space CurvesSpace Curves are inherently more difficult to draw by handthan plane Curves ; for an accurate representation we needto use instance, Figure 7 showsa computer-generated graphof the curve with parametricequationsx = (4 + sin 20t) cos ty = (4 + sin 20t) sin tz = cos 20tIt s called a toroidal spiral because it lies on a 7A toroidal spiral1414 Using Computers to Draw Space CurvesAnother interesting curve, the trefoil knot, with equationsx = (2 + cos ) cos ty = (2 + cos ) sin tz = sin graphed in Figure 8.

4 It wouldn t be easy to plot either ofthese Curves by 8A trefoil knot1515 Using Computers to Draw Space CurvesEven when a computer is used to draw a Space curve,optical illusions make it difficult to get a good impression ofwhat the curve really looks like. (This is especially true inFigure 8.)The next example shows how to cope with this 7 Use a computer to draw the curve with Vector equationr(t) = t, t2, t3 . This curve is called a twisted :We start by using the computer to plot the curve withparametric equations x = t, y = t2, z = t3 for 2 t result is shown inFigure 9(a), but it s hardto see the true nature ofthe curve from that 9(a)View of the twisted cubic1717 Example 7 SolutionMost three-dimensional computer graphing programs allowthe user to enclose a curve or surface in a box instead ofdisplaying the coordinate we look at the same curve in a box in Figure 9(b), wehave a much clearer picture of the 9(b)View of the twisted cubiccont d1818 Example 7 SolutionWe can see that it climbs from a lower corner of the box tothe upper corner nearest us, and it twists as it get an even better idea of the curve when we view itfrom different vantage 9(c) shows the result of rotating the box to giveanother 9(c)View of the twisted cubiccont d1919 Example 7 SolutionFigures 9(d), 9(e), and 9(f) show the views we get when welook directly at a face of the particular, Figure 9(d)

5 Shows the view from directlyabove the 9(d)Figure 9(e)Figure 9(f)Views of the twisted cubiccont d2020 Example 7 SolutionIt is the projection of the curve onto the xy-plane, namely,the parabola y = 9(e) shows the projection on the xz-plane, the cubiccurve z = s now obvious why the given curve is called a d2121 Using Computers to Draw Space CurvesAnother method of visualizing a Space curve is to draw it ona instance, the twisted cubic in Example 7 lies on theparabolic cylinder y = x2. (Eliminate the parameter from thefirst two parametric equations, x = t and y = t2.)Figure 10 shows both thecylinder and the twisted cubic,and we see that the curvemoves upward from the originalong the surface of the 102222 Using Computers to Draw Space CurvesWe also used this method in Example 4 to visualize thehelix lying on the circular third method for visualizing the twisted cubic is to realizethat it also lies on the cylinder z = it can be viewed as the curve of intersection of thecylinders y = x2 and z = x3.

6 (See Figure 11.)Figure 112323 Using Computers to Draw Space CurvesWe have seen that an interesting Space curve, the helix,occurs in the model of notable example of a Space curve in science is thetrajectory of a positively charged particle in orthogonallyoriented electric and magnetic fields E and Computers to Draw Space CurvesDepending on the initial velocity given the particle at theorigin, the path of the particle is either a Space curve whoseprojection on the horizontal plane is the cycloid[Figure 12(a)] or a curve whose projection is the trochoid[Figure 12(b)].Figure 12 Motion of a charged particle in orthogonallyoriented electric and magnetic fields