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1 8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in front of or behind another part of the curve. For eample, Figure shows the curve of Figure (b) as rendered b the tubeplot command in Maple. We have seen that an interesting space curve, the heli, occurs in the model of DNA. Another notable eample of a space curve in science is the trajector of a positivel charged particle in orthogonall oriented electric and magnetic fields E and B. Depending on the initial velocit given the particle at the origin, the path of the particle is either a space curve whose projection on the horiontal plane is the ccloid we studied in Section. [Figure (a)] or a curve whose projection is the trochoid investigated in Eercise 4 in Section. [Figure (b)]. B B E E (a) r(t) = kt-sin t, -cos t, tl FIGURE Motion of a charged particle in orthogonall oriented electric and magnetic fields t t (b) r(t) = kt- sin t, - cos t, tl For further details concerning the phsics involved and animations of the trajectories of the particles, see the following websites: N N FIGURE Beam/. EXERCISES Find the domain of the vector function.. r t s4 t, e t, ln t. r t t t i sin t j ln 9 t k 9. r t t, cos t, sin t.. r t, cos t, sin t.. r t t i t 4 j t 6 k 4. r t cos t i cos t j sin t k r t t, t, t r t t i tj k 6 Find the limit lim cos t, sin t, t ln t t l lim t l e t s t,, t t t lim t l e t i t sin t j cos t k lim t, e t l arctan t, ln t t 7 4 Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. 7. r t sin t, t 8. r t t, t 5 8 Find a vector equation and parametric equations for the line segment that joins P to Q. 5. P,,, 6. P,,, 7. P,,, 8. P, 4,, Q,, Q,, Q 4,, 7 Q 6,, 9 4 Match the parametric equations with the graphs (labeled I VI). Give reasons for our choices. 9. cos 4t, t,. t, t, e t sin 4t

2 CHAPTER VECTOR FUNCTIONS AND SPACE CURVES 8. t, t, t. e t cos t, e t sin t,. cos t, sin t, sin 5t 4. cos t, sin t, ln t I II e t ;. Graph the curve with parametric equations cos 6t cos t, cos 6t sin t, cos 6t. Eplain the appearance of the graph b showing that it lies on a cone. ; 4. Graph the curve with parametric equations s.5 cos t cos t s.5 cos t sin t.5 cos t III IV Eplain the appearance of the graph b showing that it lies on a sphere. 5. Show that the curve with parametric equations t, t, t passes through the points (, 4, ) and (9, 8, 8) but not through the point (4, 7, 6). V VI 6 8 Find a vector function that represents the curve of intersection of the two surfaces. 6. The clinder 4 and the surface 7. The cone s and the plane 8. The paraboloid 4 and the parabolic clinder 5. Show that the curve with parametric equations t cos t, t sin t, t lies on the cone, and use this fact to help sketch the curve. 6. Show that the curve with parametric equations sin t, cos t, sin t is the curve of intersection of the surfaces and. Use this fact to help sketch the curve. 7. At what points does the curve r t t i t t k intersect the paraboloid? 8. At what points does the heli r t sin t, cos t, t intersect the sphere 5? ; 9 Use a computer to graph the curve with the given vector equation. Make sure ou choose a parameter domain and viewpoints that reveal the true nature of the curve r t cos t sin t, sin t sin t, cos t r t t, ln t, t r t t, t sin t, t cos t. r t t, e t, cos t ; 9. Tr to sketch b hand the curve of intersection of the circular clinder 4 and the parabolic clinder. Then find parametric equations for this curve and use these equations and a computer to graph the curve. ; 4. Tr to sketch b hand the curve of intersection of the parabolic clinder and the top half of the ellipsoid Then find parametric equations for this curve and use these equations and a computer to graph the curve. 4. If two objects travel through space along two different curves, it s often important to know whether the will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given b the vector functions r t t, 7t, t for t. Do the particles collide? 4. Two particles travel along the space curves r t t, t, t Do the particles collide? Do their paths intersect? 4. Suppose u and v are vector functions that possess limits as t l a and let c be a constant. Prove the following properties of limits. (a) lim t l a r t t, 6t, 4t u t v t lim u t lim v t t l a t l a r t 4t, t, 5t 6

3 88 CHAPTER VECTOR FUNCTIONS This means that we can evaluate an integral of a vector function b integrating each component function. We can etend the Fundamental Theorem of Calculus to continuous vector functions as follows: b r t dt R t ] b a R b R a a where R is an antiderivative of r, that is, R t r t. We use the notation r t dt for indefinite integrals (antiderivatives). EXAMPLE 5 If r t cos t i sin t j t k, then r t dt cos t dt i sin t dt j t dt k sin t i cos t j t k C where C is a vector constant of integration, and r t dt [ sin t i cos t j t k] i j 4 k M. EXERCISES. The figure shows a curve C given b a vector function r t. (a) Draw the vectors r 4.5 r 4 and r 4. r 4. (b) Draw the vectors r 4.5 r 4.5 (c) Write epressions for r 4 and the unit tangent vector T(4). (d) Draw the vector T(4). and r(4.5) C r(4.) r(4) r 4. r 4. R P Q (b) Draw the vector r starting at (, ) and compare it with the vector r. r. Eplain wh these vectors are so close to each other in length and direction. 8 (a) Sketch the plane curve with the given vector equation. (b) Find r t. (c) Sketch the position vector r t and the tangent vector r t for the given value of t.. r t t, t, t 4. r t t, st, t 5. r t sin t i cos t j, 6. r t e t i e t j, 7. r t e t i e t j, t t t 4 8. r t cos t i sin t j, t 6. (a) Make a large sketch of the curve described b the vector function r t t, t, t, and draw the vectors r(), r(.), and r(.) r(). 9 6 Find the derivative of the vector function. 9. r t t sin t, t, t cos t

4 SECTION. DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS 89. r t tan t, sec t, t. r t i j e 4t k. r t sin t i s t j k. r t e t i j ln t k 4. r t at cos t i b sin t j c cos t k 5. r t a t b t c 6. r t t a b t c 7 Find the unit tangent vector T t at the point with the given value of the parameter t. 7. r t te t, arctan t, e t, t 8. r t 4st i t j t k, t 9. r t cos t i t j sin t k, t. r t sin t i cos t j tan t k,. If r t t, t, t, find r t, T, r t, and r t r t.. If r t e t, e t, te t, find T, r, and r t r t. 6 Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.. st, t t, t t; 4. e t, te t, te t ;,, 5. e t cos t, e t sin t, e t ; 6. ln t, st, t ; ; 7 9 Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate b graphing both the curve and the tangent line on a common screen. 7. t, e t, t t ; 8. cos t, sin t, 4 cos t; 9. t cos t, t, t sin t;,,,, t 4,,,,,, (s,, ). (a) Find the point of intersection of the tangent lines to the curve r t sin t, sin t, cos t at the points where t and t.5. ; (b) Illustrate b graphing the curve and both tangent lines.. The curves r t t, t, t and r t sin t, sin t, t intersect at the origin. Find their angle of intersection correct to the nearest degree.. At what point do the curves r t t, t, t and r s s, s, s intersect? Find their angle of intersection correct to the nearest degree. 8 Evaluate the integral t i 9t j 5t 4 k dt 4 t j t t k dt sin t cos t i sin t cos t j sin t cos t k dt (t i tst j t sin t k) dt e t i t j ln t k dt cos t i sin t j t k dt 9. Find r t if r t t i t j st k and r i j. 4. Find r t if r t t i e t j te t k and r i j k. 4. Prove Formula of Theorem. 4. Prove Formula of Theorem. 4. Prove Formula 5 of Theorem. 44. Prove Formula 6 of Theorem. 45. If u t sin t, cos t, t and v t t, cos t, sin t, use Formula 4 of Theorem to find 46. If u and v are the vector functions in Eercise 45, use Formula 5 of Theorem to find 47. Show that if r is a vector function such that r eists, then d 48. Find an epression for u t v t w t. dt 49. d If r t, show that r t r t. d r t r t r t r t dt r t dt r t r t r t r t [Hint: ] d u t v t dt d u t v t dt 5. If a curve has the propert that the position vector r t is alwas perpendicular to the tangent vector r t, show that the curve lies on a sphere with center the origin. 5. If u t r t r t r t, show that u t r t r t r t

5 86 CHAPTER VECTOR FUNCTIONS osculating circle = EXAMPLE 8 Find and graph the osculating circle of the parabola at the origin. SOLUTION From Eample 5 the curvature of the parabola at the origin is. So the radius of the osculating circle at the origin is and its center is (, ). Its equation is therefore ( ) 4 FIGURE 9 For the graph in Figure 9 we use parametric equations of this circle: cos t sin t We summarie here the formulas for unit tangent, unit normal and binormal vectors, and curvature. M TEC Visual.C shows how the osculating circle changes as a point moves along a curve. T t r t r t dt N t T t ds r t T t T t r t r t r t B t T t N t. EXERCISES 6 Find the length of the curve.. r t sin t, 5t, cos t,. r t t, t, t, 4. r t cos t i sin t j ln cos t k, 5. r t i t j t k, 6. r t t i 8t j t k, 7 9 Find the length of the curve correct to four decimal places. (Use our calculator to approimate the integral.) 7. r t st, t, t, t 4 8. r t t, ln t, t ln t, 9. r t sin t, cos t, tan t, t. r t st i e t j e t k, t t t t t t 4 t 4 ;. Graph the curve with parametric equations sin t, sin t, sin t. Find the total length of this curve correct to four decimal places.. Let C be the curve of intersection of the parabolic clinder and the surface. Find the eact length of C from the origin to the point 6, 8, 6.. Find, correct to four decimal places, the length of the curve of intersection of the clinder 4 4 and the plane. 4 Reparametrie the curve with respect to arc length measured from the point where t in the direction of increasing t.. r t t i t j 5 4t k 4. r t e t cos t i j e t sin t k 5. Suppose ou start at the point,, and move 5 units along the curve sin t, 4t, cos t in the positive direction. Where are ou now? 6. Reparametrie the curve r t t i t t j with respect to arc length measured from the point (, ) in the direction of increasing t. Epress the reparametriation in its simplest form. What can ou conclude about the curve? 7 (a) Find the unit tangent and unit normal vectors T t and N t. (b) Use Formula 9 to find the curvature. 7. r t sin t, 5t, cos t

6 846 CHAPTER VECTOR FUNCTIONS where c c. Then r r v h GM c cos GM r v h e cos where e c GM. But r v h r v h h h h h where h h. So r h GM e cos eh c e cos Writing d h c, we obtain the equation r ed e cos Comparing with Theorem.6.6, we see that Equation is the polar equation of a conic section with focus at the origin and eccentricit e. We know that the orbit of a planet is a closed curve and so the conic must be an ellipse. This completes the derivation of Kepler s First Law. We will guide ou through the derivation of the Second and Third Laws in the Applied Project on page 848. The proofs of these three laws show that the methods of this chapter provide a powerful tool for describing some of the laws of nature..4 EXERCISES. The table gives coordinates of a particle moving through space along a smooth curve. (a) Find the average velocities over the time intervals [, ], [.5, ], [, ], and [,.5]. (b) Estimate the velocit and speed of the particle at t. (d) Draw an approimation to the vector v() and estimate the speed of the particle at t. t r(.4) r() r(.5). The figure shows the path of a particle that moves with position vector r t at time t. (a) Draw a vector that represents the average velocit of the particle over the time interval t.4. (b) Draw a vector that represents the average velocit over the time interval.5 t. (c) Write an epression for the velocit vector v(). 8 Find the velocit, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocit and acceleration vectors for the specified value of t. r t t, t., t 4. r t t, 4st, t

7 SECTION.4 MOTION IN SPACE: VELOCITY AND ACCELERATION r t cos t i sin t j, 6. r t e t i e t j, 7. r t t i t j k, 8. r t t i cos t j sin t k, 9 4 Find the velocit, acceleration, and speed of a particle with the given position function. 9. r t t, t, t. r t cos t, t, sin t. r t s t i e t j e t k. r t t i ln t j t k. r t e t cos t i sin t j t k 4. t 5 6 Find the velocit and position vectors of a particle that has the given acceleration and the given initial velocit and position. 5. a t i j, v k, t r t t sin t i t cos t j t k t t r i 6. a t i 6t j t k, v i, 7 8 (a) Find the position vector of a particle that has the given acceleration and the specified initial velocit and position. ; (b) Use a computer to graph the path of the particle. 7. a t t i sin t j cos t k, v i, 8. a t t i e t j e t k, v k, r j k r j r j k 9. The position function of a particle is given b r t t, 5t, t 6t. When is the speed a minimum?. What force is required so that a particle of mass m has the position function r t t i t j t k?. A force with magnitude N acts directl upward from the -plane on an object with mass 4 kg. The object starts at the origin with initial velocit v i j. Find its position function and its speed at time t.. Show that if a particle moves with constant speed, then the velocit and acceleration vectors are orthogonal.. A projectile is fired with an initial speed of 5 m s and angle of elevation. Find (a) the range of the projectile, (b) the maimum height reached, and (c) the speed at impact. 4. Rework Eercise if the projectile is fired from a position m above the ground. 5. A ball is thrown at an angle of 45 to the ground. If the ball lands 9 m awa, what was the initial speed of the ball? 6. A gun is fired with angle of elevation. What is the mule speed if the maimum height of the shell is 5 m? 7. A gun has mule speed 5 m s. Find two angles of elevation that can be used to hit a target 8 m awa. 8. A batter hits a baseball ft above the ground toward the center field fence, which is ft high and 4 ft from home plate. The ball leaves the bat with speed 5 ft s at an angle 5 above the horiontal. Is it a home run? (In other words, does the ball clear the fence?) 9. A medieval cit has the shape of a square and is protected b walls with length 5 m and height 5 m. You are the commander of an attacking arm and the closest ou can get to the wall is m. Your plan is to set fire to the cit b catapulting heated rocks over the wall (with an initial speed of 8 m s). At what range of angles should ou tell our men to set the catapult? (Assume the path of the rocks is perpendicular to the wall.). A ball with mass.8 kg is thrown southward into the air with a speed of m s at an angle of to the ground. A west wind applies a stead force of 4 N to the ball in an easterl direction. Where does the ball land and with what speed? ;. Water traveling along a straight portion of a river normall flows fastest in the middle, and the speed slows to almost ero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 4 m apart. If the maimum water speed is m s, we can use a quadratic function as a basic model for the rate of water flow units from the west bank: f 4 4. (a) A boat proceeds at a constant speed of 5 m s from a point A on the west bank while maintaining a heading perpendicular to the bank. How far down the river on the opposite bank will the boat touch shore? Graph the path of the boat. (b) Suppose we would like to pilot the boat to land at the point B on the east bank directl opposite A. If we maintain a constant speed of 5 m s and a constant heading, find the angle at which the boat should head. Then graph the actual path the boat follows. Does the path seem realistic?. Another reasonable model for the water speed of the river in Eercise is a sine function: f sin 4. If a boater would like to cross the river from A to B with constant heading and a constant speed of 5 m s, determine the angle at which the boat should head. 8 Find the tangential and normal components of the acceleration vector.. r t t t i t j 4. r t t i t t j 5. r t cos t i sin t j t k 6. r t t i t j t k

8 A4 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES Hperboloid of two sheets 5. Ellipsoid (,, ) _4 _4 4 4 _4 _ (,, ) (,, ) (,, ) 4. = (,, _) =œ + PROBLEMS PLUS N PAGE , paraboloid 47. (a) (b) Circle (c) Ellipse 5.. (s.5) m. (a) c c c c c (b) t, t (c) 4 CHAPTER EXERCISES. PAGE 8 N.,.,, 5. i j k π (,, ) CHAPTER REVIEW N PAGE 8 (π,, ) True-False Qui. True. True 5. True 7. True 9. True. False. False 5. False 7. True Eercises. (a) 69 (b) 68, (c) Center 4,,, radius 5. u v s; u v s; out of the page 5., 4 7. (a) (b) (c) (d) 9. cos ( ) 7. (a) 4,, 4 (b) s4. 66 N, 4 N 5. 4 t, t, t 7. t, t, 4 5t (,4,4). Skew s6 9. Plane. Cone.. = 5. r t t, t, t, t ; t, t, t, t 7. r t t, t, 5t, t ; t, t, 5t, t 9. VI. IV. V 5. 7.,,,,,

9 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES A5 9. r 4 h r 4 (c) r 4 lim ; T 4 h l h r 4 r 4. (a), (c) (b) r t, t (_, ). _ rª(_) r(_) 5. (a), (c) 7. (a), (c) œ, œ π rª 4 π r 4 r() (, ) rª(). _ 7. r t t i t j t k 9. cos t, sin t, 4 cos t 4. Yes EXERCISES.. (a) (b), (d) _ N PAGE 88 r(4.5) C r(4.) r(4) r(4.5)-r(4).5 r(4.5) C R r(4.) r(4.)-r(4). R Q r(4.5)-r(4) Q r(4.)-r(4) P T(4) _ (b) r t cos t i sin t j (b) r t e t i e t j 9. r t t cos t sin t, t, cos t t sin t. r t 4e 4t k. r t te t i [ t k 5. r t b tc 7.,, 9. 5 j 4 5 k., t, t, s4, s4, s4,,, 6t, 6t, 6t,. t, t, 4t 5. t, t, t 7. t, t, t 9. t, t, t i j 5 k 5. i j k 7. e t i t j t ln t t k C t i t j ( t ) k t cos t sin t cos t sin t EXERCISES. N PAGE 86. s9. e e r t s s9 s i s9 s j 5 4 s9 s k 5. sin, 4, cos 7. (a) ( s9) cos t, 5 s9, ( s9) sin t, sin t,, cos t (b) 9 9. (a) e t e t, se t, se t e t set, e t,, (b). se t e t 4 4t s / 9. 5s. ( ln, s) ; approaches. (a) P (b).,.7 5. =k() 4 r(4) P 7. a is f, b is _4 4 _

10 A6 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES 9. k(t) t 6s4 cos t cos t. s i e t j e t k, e t j e t k, e t e t 7 cos t. e t cos t sin t i sin t cos t j t k, e t sin t i cos t j t k, e t st t 5. v t t i t j k, r t ( t i t j t k 7. (a) (b) integer multiples of r t ( t t) i t sin t j ( 4 4 cos t) k π 4π 6π 4. (se t ) , ( 5 ) 8 4, ( 5 ) t,,,,,,,, t 4. r t t i t j 5 t k,. (a) km (b). km (c) 5 m s 5. m s 7.., , (a) 6 m (b).6 upstream.6.4. v t s5t ,, 57. t 4 4t Å m _4 4 _ EXERCISES.4 N PAGE 846. (a).8i.8j.7k,.i.4j.6k,.8i.8j.k,.8i.8j.4k (b).4i.8j.5k,.58. v t t, v() a t, (_, ) v t st a(). 6t, 6 5., 7. e t e t, s cm s, 9. cm s 4. t CHAPTER REVIEW N PAGE 85 True-False Qui. True. False 5. False 7. True 9. False. True Eercises. (a) 5. v t sin t i cos t j a t cos t i sin t j v t s5 sin t 4 (, ) π v π a, œ (, ) (,, ) (,, ) 7. v t i t j a t j v t s 4t (,, ) 9. t, t, t,, 6t,, t s9t 8 a() v() (b) r t i sin t j cos t k, r t cos t j sin t k. r t 4 cos t i 4 sin t j 5 4 cos t k, t 5. i j k (a) t, t, st 4 t (b) t, t 4, t t st 8 4t 6 t 4 5t (c) st 8 4t 6 t 4 5t t 4 t v t ln t i j e t k 7., v t s ln t ln t e t, a t t i e t k

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:

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