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

Transcription

1 8 CHAPTER 3 VECTOR FUNCTIONS N Some computer algebra sstems provie 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 behin another part of the curve. For eample, Figure 3 shows the curve of Figure (b) as renere b the tubeplot comman in Maple. We have seen that an interesting space curve, the heli, occurs in the moel of DNA. Another notable eample of a space curve in science is the trajector of a positivel charge particle in orthogonall oriente electric an magnetic fiels E an B. Depening 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 ccloi we stuie in Section. [Figure (a)] or a curve whose projection is the trochoi investigate in Eercise 4 in Section. [Figure (b)]. B B E E (a) r(t) = kt-sin t, -cos t, tl FIGURE Motion of a charge particle in orthogonall oriente electric an magnetic fiels t t 3 3 (b) r(t) = kt- sin t, - cos t, tl For further etails concerning the phsics involve an animations of the trajectories of the particles, see the following websites: N N FIGURE Beam/ 3. EXERCISES Fin the omain of the vector function.. rt s4 t, e 3t, lnt. rt t t i sin t j ln9 t k 9. rt t, cos t, sin t.. rt, cos t, sin t. 3. rt t i t 4 j t 6 k 4. rt cos t i cos t j sin t k rt t, 3t, t rt t i tj k 3 6 Fin the limit lim cos t, sin t, t ln t t l lim t l e t s t 3,, t t t lim t l e 3t 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. Inicate with an arrow the irection in which t increases. 7. rt sin t, t 8. rt t 3, t 5 8 Fin a vector equation an parametric equations for the line segment that joins P to Q. 5. P,,, 6. P,,, 7. P,,, 8. P, 4,, Q,, 3 Q, 3, Q4,, 7 Q6,, 9 4 Match the parametric equations with the graphs (labele I VI). Give reasons for our choices. 9. cos 4t, t,. t, t, e t sin 4t

2 CHAPTER 3 VECTOR FUNCTIONS AND SPACE CURVES 83. t, t, t. e t cos t, e t sin t, 3. cos t, sin t, sin 5t 4. cos t, sin t, ln t I II e t ; 33. 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. ; 34. 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. 35. Show that the curve with parametric equations t, 3t, t 3 passes through the points (, 4, ) an (9, 8, 8) but not through the point (4, 7, 6). V VI Fin a vector function that represents the curve of intersection of the two surfaces. 36. The cliner 4 an the surface 37. The cone s an the plane 38. The paraboloi 4 an the parabolic cliner 5. Show that the curve with parametric equations t cos t, t sin t, t lies on the cone, an 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 an. Use this fact to help sketch the curve. 7. At what points oes the curve rt t i t t k intersect the paraboloi? 8. At what points oes the heli rt sin t, cos t, t intersect the sphere 5? ; 9 3 Use a computer to graph the curve with the given vector equation. Make sure ou choose a parameter omain an viewpoints that reveal the true nature of the curve rt cos t sin t, sin t sin t, cos t rt t, ln t, t rt t, t sin t, t cos t 3. rt t, e t, cos t ; 39. Tr to sketch b han the curve of intersection of the circular cliner 4 an the parabolic cliner. Then fin parametric equations for this curve an use these equations an a computer to graph the curve. ; 4. Tr to sketch b han the curve of intersection of the parabolic cliner an the top half of the ellipsoi Then fin parametric equations for this curve an use these equations an a computer to graph the curve. 4. If two objects travel through space along two ifferent curves, it s often important to know whether the will collie. (Will a missile hit its moving target? Will two aircraft collie?) The curves might intersect, but we nee 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 collie? 4. Two particles travel along the space curves r t t, t, t 3 Do the particles collie? Do their paths intersect? 43. Suppose u an v are vector functions that possess limits as t l a an let c be a constant. Prove the following properties of limits. (a) lim t l a r t t, 6t, 4t ut vt lim ut lim vt t l a t l a r t 4t 3, t, 5t 6

3 88 CHAPTER 3 VECTOR FUNCTIONS This means that we can evaluate an integral of a vector function b integrating each component function. We can eten the Funamental Theorem of Calculus to continuous vector functions as follows: b rt t Rt] b a Rb Ra a where R is an antierivative of r, that is, Rt rt. We use the notation rt t for inefinite integrals (antierivatives). EXAMPLE 5 If rt cos t i sin t j t k, then rt t cos t t i sin t t j t t k sin t i cos t j t k C where C is a vector constant of integration, an rt t [ sin t i cos t j t k] i j 4 k M 3. EXERCISES. The figure shows a curve C given b a vector function rt. (a) Draw the vectors r4.5 r4 an r4. r4. (b) Draw the vectors r4.5 r4.5 (c) Write epressions for r4 an the unit tangent vector T(4). () Draw the vector T(4). an r(4.5) C r(4.) r(4) r4. r4. R P Q (b) Draw the vector r starting at (, ) an compare it with the vector r. r. Eplain wh these vectors are so close to each other in length an irection. 3 8 (a) Sketch the plane curve with the given vector equation. (b) Fin rt. (c) Sketch the position vector rt an the tangent vector rt for the given value of t. 3. rt t, t, t 4. rt t, st, t 5. rt sin t i cos t j, 6. rt e t i e t j, 7. rt e t i e 3t j, t t t 4 8. rt cos t i sin t j, t 6. (a) Make a large sketch of the curve escribe b the vector function rt t, t, t, an raw the vectors r(), r(.), an r(.) r(). 9 6 Fin the erivative of the vector function. 9. rt t sin t, t, t cos t

4 SECTION 3. DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS 89. rt tan t, sec t, t. rt i j e 4t k. rt sin t i s t j k 3. rt e t i j ln 3t k 4. rt at cos 3t i b sin 3 t j c cos 3 t k 5. rt a t b t c 6. rt t a b t c 7 Fin the unit tangent vector Tt at the point with the given value of the parameter t. 7. rt te t, arctan t, e t, t 8. rt 4st i t j t k, t 9. rt cos t i 3t j sin t k, t. rt sin t i cos t j tan t k,. If rt t, t, t 3, fin rt, T, rt, an rt rt.. If rt e t, e t, te t, fin T, r, an rt rt. 3 6 Fin parametric equations for the tangent line to the curve with the given parametric equations at the specifie point. 3. st, t 3 t, t 3 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 Fin parametric equations for the tangent line to the curve with the given parametric equations at the specifie point. Illustrate b graphing both the curve an 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,, 3,,,, (s3,, ) 3. (a) Fin the point of intersection of the tangent lines to the curve rt sin t, sin t, cos t at the points where t an t.5. ; (b) Illustrate b graphing the curve an both tangent lines. 3. The curves r t t, t, t 3 an r t sin t, sin t, t intersect at the origin. Fin their angle of intersection correct to the nearest egree. 3. At what point o the curves r t t, t, 3 t an r s 3 s, s, s intersect? Fin their angle of intersection correct to the nearest egree Evaluate the integral t 3 i 9t j 5t 4 k t 4 t j t t k t 3 sin t cos t i 3 sin t cos t j sin t cos t k t (t i tst j t sin t k) t e t i t j ln t k t cos t i sin t j t k t 39. Fin rt if rt t i 3t j st k an r i j. 4. Fin rt if rt t i e t j te t k an r i j k. 4. Prove Formula of Theorem Prove Formula 3 of Theorem Prove Formula 5 of Theorem Prove Formula 6 of Theorem If ut sin t, cos t, t an vt t, cos t, sin t, use Formula 4 of Theorem 3 to fin 46. If u an v are the vector functions in Eercise 45, use Formula 5 of Theorem 3 to fin 47. Show that if r is a vector function such that r eists, then 48. Fin an epression for ut vt wt. t 49. If rt, show that rt rt. rt rt rt rt t rt t rt rt rt rt [Hint: ] ut vt t ut vt t 5. If a curve has the propert that the position vector rt is alwas perpenicular to the tangent vector rt, show that the curve lies on a sphere with center the origin. 5. If ut rt rt rt, show that ut rt rt rt

5 836 CHAPTER 3 VECTOR FUNCTIONS osculating circle = EXAMPLE 8 Fin an 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 raius of the osculating circle at the origin is an 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 an binormal vectors, an curvature. M TEC Visual 3.3C shows how the osculating circle changes as a point moves along a curve. Tt rt rt T Nt Tt s rt Tt Tt rt rt rt 3 Bt Tt Nt 3.3 EXERCISES 6 Fin the length of the curve.. rt sin t, 5t, cos t,. rt t, t, 3 t 3, 4. rt cos t i sin t j ln cos t k, 5. rt i t j t 3 k, 6. rt t i 8t 3 j 3t k, 7 9 Fin the length of the curve correct to four ecimal places. (Use our calculator to approimate the integral.) 7. rt st, t, t, t 4 8. rt t, ln t, t ln t, 9. rt sin t, cos t, tan t, t 3. rt 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 3t. Fin the total length of this curve correct to four ecimal places.. Let C be the curve of intersection of the parabolic cliner an the surface 3. Fin the eact length of C from the origin to the point 6, 8, 36.. Fin, correct to four ecimal places, the length of the curve of intersection of the cliner 4 4 an the plane. 3 4 Reparametrie the curve with respect to arc length measure from the point where t in the irection of increasing t. 3. rt t i 3t j 5 4t k 4. rt e t cos t i j e t sin t k 5. Suppose ou start at the point,, 3 an move 5 units along the curve 3 sin t, 4t, 3 cos t in the positive irection. Where are ou now? 6. Reparametrie the curve rt t i t t j with respect to arc length measure from the point (, ) in the irection of increasing t. Epress the reparametriation in its simplest form. What can ou conclue about the curve? 7 (a) Fin the unit tangent an unit normal vectors Tt an Nt. (b) Use Formula 9 to fin the curvature. 7. rt sin t, 5t, cos t

6 A4 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES Hperboloi of two sheets 35. Ellipsoi (,, ) _4 _4 4 4 _4 _ (,, ) (,, ) (,, ) 4. = (,, _) =œ + PROBLEMS PLUS N PAGE , paraboloi 47. (a) (b) Circle (c) Ellipse 5.. (s3.5) m 3. (a) c cc cc (b) t, t (c) 43 CHAPTER 3 EXERCISES 3. PAGE 8 N., 3.,, 5. i j k π (,, ) CHAPTER REVIEW N PAGE 8 (π,, ) True-False Qui. True 3. True 5. True 7. True 9. True. False 3. False 5. False 7. True Eercises. (a) 69 (b) 68, (c) Center 4,, 3, raius 5 3. u v 3s; u v 3s; out of the page 5., 4 7. (a) (b) (c) () 9. cos ( 3 ) 7. (a) 4, 3, 4 (b) s N, 4 N t, t, 3t 7. t, t, 4 5t (,4,4) 3. Skew s6 9. Plane 3. Cone. 3. = 5. rt t, t, 3t, t ; t, t, 3t, t 7. rt 3t, t, 5t, t ; 3t, t, 5t, t 9. VI. IV 3. V 5. 7.,,,,,

7 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES A5 9. r4 h r4 (c) r4 lim ; T4 h l h r4 r4 3. (a), (c) (b) rt, t (_3, ) 3. _ rª(_) r(_) 5. (a), (c) 7. (a), (c) œ, œ π rª 4 π r 4 r() (, ) rª() 33. _ 37. rt t i t j t k 39. cos t, sin t, 4 cos t 4. Yes EXERCISES 3.. (a) (b), () _ 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) rt cos t i sin t j (b) rt e t i 3e 3t j 9. rt t cos t sin t, t, cos t t sin t. rt 4e 4t k 3. rt te t i [3 3t k 3 5. rt b tc 7. 3, 3, 9. 5 j k., t, 3t, s4, s4, 3s4,,, 6t, 6t, 6t, 3. 3 t, t, 4t 5. t, t, t 7. t, t, t 9. t, t, t i 3 j 5 k 35. i j k 37. e t i t j t ln t tk C t i t 3 j ( 3t 3 3) k t cos t sin t cos t sin t EXERCISES 3.3 N PAGE 836. s9 3. e e rts s9 s i 3 s9 s j 5 4 s9 s k 5. 3 sin, 4, 3 cos 7. (a) (s9) cos t, 5s9, (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 33. (a) P (b).3, =k() 4 r(4) P 37. a is f, b is _4 4 _