Vector Functions - Whitman College

1 13 Vector eCurvesWe have already seen that a convenient way to describe a line in three dimensions is toprovide a Vector that points to every point on the line as a parametertvaries, likeh1,2,3i+th1, 2,2i=h1 +t,2 2t,3 + that this gives a particularly simple geometric object, there is nothing special aboutthe individual Functions oftthat make up the coordinates of this Vector any Vector witha parameter, likehf(t), g(t), h(t)i, will describe some curve in three dimensions astvariesthrough all possible the curveshcost,sint,0i,hcost,sint, ti, andhcost,sint, , the first two coordinates in all three Functions trace out the points on theunit circle, starting with (1,0) whent= 0 and proceeding counter-clockwise around thecircle astincreases.

2 In the first case, thezcoordinate is always 0, so this describes preciselythe unit circle in thex-yplane. In the second case, thexandycoordinates still describea circle, but now thezcoordinate varies, so that the height of the curve matches the valueoft. Whent= , for example, the resulting Vector ish 1,0, i. A bit of thought shouldconvince you that the result is a helix. In the third Vector , thezcoordinate varies twice asfast as the parametert, so we get a stretched out helix. Both are shown in figure the left is the first helix, shown fortbetween 0 and 4 ; on the right is the second helix,shown fortbetween 0 and 2.

3 Both start and end at the same point, but the first helixtakes two full turns to get there, because itszcoordinate grows more 13 Vector FunctionsFigure helixes. (AP)A Vector expression of the formhf(t), g(t), h(t)iis called avector function; it isa function from the real numbersRto the set of all three-dimensional vectors. We canalternately think of it as three separate Functions ,x=f(t),y=g(t), andz=h(t), thatdescribe points in space. In this case we usually refer to theset of equations asparametricequationsfor the curve, just as for a line. While the parametertin a Vector functionmight represent any one of a number of physical quantities, or be simply a pure number ,it is often convenient and useful to think oftas representing time.

4 The Vector functionthen tells you where in space a particular object is at any Functions can be difficult to understand, that is, difficult to picture. Whenavailable, computer software can be very helpful. When working by hand, one usefulapproach is to consider the projections of the curve onto the three standard coordinateplanes. We have already done this in part: in example we noted that all three curvesproject to a circle in thex-yplane, sincehcost,sintiis a two dimensional Vector functionfor the unit the projections ofhcost,sint,2tionto thex-zplane and they-zplane. The two dimensional Vector function for the projection onto thex-zplane ishcost,2ti, or in parametric form,x= cost,z= 2t.

5 By eliminatingtwe get the equationx= cos(z/2), the familiar curve shown on the left in figure For the projection ontothey-zplane, we start with the Vector functionhsint,2ti, which is the same asy= sint,z= 2t. Eliminatingtgivesy= sin(z/2), as shown on the right in figure the curver=hsint,cost,cos the curver=htcost, tsint, Calculus with Vector functions3312 4 4 projections ofhcost,sint,2tionto the curver=ht, t2, the curver=hcos(20t) 1 t2,sin(20t) 1 t2, a Vector function for the curve of intersection ofx2+y2= 9 andy+z= 2. bug is crawling outward along the spoke of a wheel that lies along a radius of the bug is crawling at 1 unit per second and the wheel is rotating at 1 radian per the wheel lies in they-zplane with center at the origin, and at timet= 0 the spokelies along the positiveyaxis and the bug is at the origin.

6 Find a Vector functionr(t) for theposition of the bug at timet. is the difference between the parametric curvesf(t) =ht, t, t2i,g(t) =ht2, t2, t4i, andh(t) =hsin(t),sin(t),sin2(t)iastruns over all real numbers? each of the curves below in 2 dimensions, projected onto each of the three standardplanes (thex-y,x-z, andy-zplanes). (t) =ht, t3, t2i,tranges over all real (t) =ht2, t 1, t2+ 5ifor 0 t pointsA= (a1, a2, a3) andB= (b1, b2, b3), give parametric equations for the linesegmentconnectingAandB. Be sure to give a parametric plot and a set oftvalues, we can associate a direction . For example,the curvehcost,sintiis the unit circle traced counterclockwise.

7 How can we amend a setofgiven parametric equations andtvalues to get the same curve, only traced backwards? uluswithve torfun tionsA Vector functionr(t) =hf(t), g(t), h(t)iis a function of one variable that is, there isonly one input value. What makes Vector Functions more complicated than the functionsy=f(x) that we studied in the first part of this book is of course thatthe output valuesare now three-dimensional vectors instead of simply numbers. It is natural to wonder ifthere is a corresponding notion of derivative for Vector Functions . In the simpler case of332 Chapter 13 Vector Functionsa functiony=s(t), in whichtrepresents time ands(t) is position on a line, we haveseen that the derivatives (t) represents velocity; we might hope that in a similar way thederivative of a Vector function would tell us something about the velocity of an objectmoving in three way to approach the question of the derivative for vectorfunctions is to writedown an expression that is analogous to the derivative we already understand, and see ifwe can make sense of it.

8 This gives usr (t) = lim t 0r(t+ t) r(t) t= lim t 0hf(t+ t) f(t), g(t+ t) g(t), h(t+ t) h(t)i t= lim t 0hf(t+ t) f(t) t,g(t+ t) g(t) t,h(t+ t) h(t) ti=hf (t), g (t), h (t)i,if we say that what we mean by the limit of a Vector is the vectorof the individualcoordinate limits. So starting with a familiar expression for what appears to be a derivative,we find that we can make good computational sense out of it butwhat does it actuallymean?We know how to interpretr(t+ t) andr(t) they are vectors that point to locationsin space; iftis time, we can think of these points as positions of a moving object at timesthat are tapart.

9 We also know what r=r(t+ t) r(t) means it is a Vector thatpoints from the head ofr(t) to the head ofr(t+ t), assuming both have their tails atthe origin. So when tis small, ris a tiny Vector pointing from one point on the pathof the object to a nearby point. As tgets close to 0, this Vector points in a directionthat is closer and closer to the direction in which the objectis moving; geometrically, itapproaches a Vector tangent to the path of the object at a particular (t) rr(t+ t)..Figure the , the Vector rapproaches 0 in length; the vectorh0,0,0iis not veryinformative. By dividing by t, when it is small, we effectively keep magnifying the Calculus with Vector functions333of rso that in the limit it doesn t disappear.

10 Thus the limiting vectorhf (t), g (t), h (t)iwill (usually) be a good, non-zero Vector that is tangent to the about the length of this Vector ? It s nice that we ve kept it away from zero,but what does it measure, if anything? Consider the length ofone of the vectors thatapproaches the tangent Vector : r(t+ t) r(t) t =|r(t+ t) r(t)|| t|The numerator is the length of the Vector that points from oneposition of the object to a nearby position; this length is approximately the distance traveled by the object betweentimestandt+ t. Dividing this distance by the length of time it takes to travel thatdistance gives the average speed.