Vectors - Mathematics
8-68-6Vectors1. PlanObjectives12To describe vectorsTo solve problems that involvevector additionExamples12345Describing a VectorDescribing a Vector DirectionReal-World ConnectionAdding VectorsReal-World ConnectionCheck Skills You’ll NeedWhat You’ll Learn To describe vectors To solve problems thatGO for HelpLesson 8-12x Algebra Find the value of x. Leave your answers in simplest radical form.1.2.3. 10 "656 "134 "41involve vector additionxx16. . . And Why7To use vectors to describe thedistance and direction of anairplane flight, as inExample 3x407020New Vocabulary vector magnitude initial point terminal point resultantMath BackgroundScientific descriptions need tobe precise and concise. Becausevectors describe quantities withboth magnitude and direction,they are especially useful inscience. For example, the studyof physics employs vectorsextensively to describe forceand velocity.1Describing VectorsA vector is any quantity with magnitude (size) anddirection. There are many models for a vector.WYou can use an arrow for a vector as shown by thevelocity vector KW in the photo. The magnitudecorresponds to the distance from initial point K tothe terminal point W. The direction correspondsto the direction in which the arrow points.More Math Background: p. 414DWMagnitude25 mi/hKYou can also use an ordered pair kx, yl in thecoordinate plane for a vector. The magnitude anddirection of the vector correspond to the distanceand direction of kx, yl from the origin.Lesson Planning andResourcesSee p. 414E for a list of theresources that support this lesson.1EXAMPLEDescribing a VectorWCoordinate Geometry Describe OL asan ordered pair. Give the coordinates tothe nearest tenth.PowerPointBell Ringer PracticeOyxx50 y65Check Skills You’ll NeedFor intervention, direct students to:Using the Pythagorean TheoremUse the sine and cosine ratiosto find the values of x and y.Lesson 8-1: Example 2Extra Skills, Word Problems, ProofPractice, Ch. 8xcos 508 65x 65(cos 508) Ly41.78119463sin 508 65y 65(sin 508) 49.7928888Use sine and cosine.Solve for the variable.Use a calculator.WL is in the fourth quadrant so the y-coordinate is negative. OL k41.8, -49.8l.452Chapter 8 Right Triangles and TrigonometrySpecial NeedsBelow LevelL1In Example 1, help students understand that thex-coordinate in 41.8, 49.8 is positive and they-coordinate is negative because the direction fromthe origin to point L is right and down.452learning style: verbalL2Have students use centimeter graph paper to confirmthe distance in Example 3 and use rulers andprotractors to compare methods of describingthe vector.learning style: tactile
Quick CheckReal-World1 Describe the vector at the right as an ordered pair.Give the coordinates to the nearest tenth. –21.6, 46.2 5165 O2. TeachyxGuided InstructionConnectionA velocity vector for a “bullettrain” can have magnitude275 km/h paired with anydirection point on a compass.In many applications of vectors, you use the compass directions north, south, east,and west to describe the direction of a vector.2EXAMPLEDescribing a Vector DirectionUse compass directions to describe the direction of each vector.a.b.NW25 NEW258 south of eastQuick Check30 WEyS2 a. Sketch a vector that has the direction 308 west of north.b. Critical Thinking Give a second description for the direction of this vector.2b. 60 north of west3EXAMPLEReal-WorldN-40Wd !2225d Ex -25402 Use compass directions todescribe the direction of thevector.NWES22 Simplify.SUse a calculator to find the square root.3 A boat sailed 12 mi east and9 mi south. The trip can bedescribed by the vector 12, -9 .Use distance and direction todescribe this vector a second way.The boat sailed 15 mi at about37 south of east.To find the direction of the flight, find the angle of the vector south of west.TAN-1 2580M22 east of south47.169906tan x8 2540x tan-1 Q 2540 RxOcTo find the distance, use the Distance Formula:d #(240 2 0) 2 1 (225 2 0) 2d !1600 1 625yx40 –61.3, –51.4 ConnectionAviation An airplane lands 40 km west and25 km south from where it took off. The result ofthe trip can be described by the vector k-40, -25l.Use distance (for magnitude) and direction todescribe this vector a second way.Find the tangent ratio.Use the inverse of tangent.32.005383Use a calculator.The airplane flew about 47 km at 328 south of west.3 A small airplane lands at a point 246 mi east and 76 mi north of the point fromwhich it took off. Describe the magnitude and the direction of its flight vector.about 257 mi at 17 N of ELesson 8-6 VectorsAdvanced LearnersW1 Describe OM as an orderedpair. Give coordinates to thenearest tenth.358 east of northExample 3 shows how to describe a vector’s magnitude and direction when you aregiven its description as an ordered pair.Quick CheckIn finding 35 east of north,students should focus first ondue north and then move 35 east. Encourage students to usea compass diagram when findingvector directions.Additional ExamplesESSTeaching TipEXAMPLEPowerPointN2a.35 2453English Language Learners ELLL4After learning how to add vectors, students caninvestigate whether vector addition is commutativeand associative.learning style: verbalHelp students distinguish between south of east andeast of south. In English, adjectives mostly precede thenouns they modify, such as white house. The firstcompass direction modifies the second primarydirection.learning style: verbal453
Guided Instruction21Adding VectorsTechnology TipYou can also use a single lowercase letter,such as u , to name a vector.Have students check to seewhether their calculatorsperform vector addition.4WThis map shows vectors representing a flightfrom Houston to Memphis with a stopoverin New Orleans. The vector from Houstonto Memphis is called the sum, or resultant,of the other two vectors. You write this asConnectionto PhysicsEXAMPLEPoint out that vectors are usedextensively in physics to find theresultant of several velocities,accelerations, or forces.5WKey ConceptsProperty4Additional ExamplesWWWare shown below. Write s , theirsum, as an ordered pair.4 yEXAMPLEv2 24WAdding VectorsAdding VectorsVectors aW k4, 3l and cW k-1, 2l are shown in the diagram.Write the sum of the two vectors as an ordered pair.Then draw eW , the sum of aW and cW.aW cW k4, 3l k-1, 2l4 Vectors v 4, 3 and w 4, -3 2New OrleansuFor aW kx1, y1l and cW kx2, y2l, aW cW kx1 x2, y1 y2l.PowerPointxHoustonvYou can add vectors by adding their coordinates. You can also show the sumgeometrically.Point out that this exampleuses the Pythagorean Theorem,whereas Example 3 used theDistance Formula. Have studentscompare the two approaches.2ww u v.Alternative MethodEXAMPLEWMemphisFor: Adding Vectors ActivityUse: Interactive Textbook, 8-6wWs 8, 0 Quick CheckAdd the coordinates.Simplify.Draw a with its initial point at the origin. Thendraw c with its initial point at the terminal pointof a . Finally, draw the resultant e from the initialpoint of a to the terminal point of c .2WWW-2Wac 2 O4Wy2k3, 5l is the resultant.W 4 k4 (-1), 3 2l k3, 5l4Ox24ycea2x44 Write the sum of the two vectors k2, 3l and k-4, -2l as an ordered pair. –2, 1 5 An airplane’s speed is 250 mi/hin still air. The wind is blowing dueeast at 20 mi/h. If the airplaneheads due north, what is itsresultant speed and direction?Round answers to the nearest unit.251 mi/h, 5 east of northA canoe travelingin this directionand at this speed . . .hits thiscurrent . . .Resources Daily Notetaking Guide 8-6 L3 Daily Notetaking Guide 8-6—L1Adapted Instructionand ends uptraveling in thisdirection and atthis speed.A vector sum can show the resultof vectors that occur in sequence,such as in the airplane flightdescribed above.A vector sum can also show theresult of vectors that act at thesame time, such as when you rowin a direction different from thatof the current. See diagram at left.The velocity of the canoe is the vector sum ofthe velocities of the paddlers and the stream.ClosureSketch a vector with magnitude50 and direction 30 west of north.Describe it as an ordered pairwith coordinates rounded tothe nearest tenth. –25, 43.3 ;check that vectors are drawnfrom (0, 0) to (–25, 43.3).454454Chapter 8 Right Triangles and Trigonometry7.8.N9.NN20 WEW45 EWE50 SSS
5EXAMPLEReal-World3. PracticeConnectionNavigation A ferry shuttles people fromone side of a river to the other. The speedof the ferry in still water is 25 mi/h. Theriver flows directly south at 7 mi/h. If theferry heads directly west, what are theferry’s resultant speed and direction?25 mi/hx Nc7 mi/hWAssignment Guide1 A B 1-16, 29, 30, 32, 33,ES40, 45, 46The diagram shows the sum of the two vectors. To find the ferry’s resultant speed,use the Pythagorean Theorem.c2 252 72The lengths of the legs are 25 and 7.c2 674Simplify.25.961510c Real-WorldConnection7tan x8 257Rx tan-1 Q 25Use a calculator.To check students’ understandingof key skills and concepts, go overExercises 2, 18, 31, 35, 40.Use the inverse of the tangent.Use a calculator.Error Prevention!The ferry’s speed is about 26 mi/h. Its direction is about 168 south of west.Quick Check5 Critical Thinking Use the diagram to find the angleat which the ferry must head upriver in order to traveldirectly across the river.about 16 north of westExercises 2, 3 Students mayforget to determine the signs ofthe coordinates. Remind studentsto check which quadrant containsthe vector in the diagram.25 mi/h7 mi/hx Auditory LearnersEXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.Practice and Problem SolvingAPractice by ExampleDescribe each vector as an ordered pair. Give the coordinates to the nearest tenth.1.Example 1GO forHelp53-5556-60Homework Quick CheckUse the tangent ratio.1 5 . 6 4 224 6x 17-28, 31, 34-39,41-44, 47, 48C Challenge49-52Test PrepMixed ReviewTo find the ferry’s resultant direction, use trigonometry.Ferry service is essential inremote regions such as on theMackenzie River in Canada’sNorthwest Territories.2 A B2.y10 (page 452)312900O yxExercises 4–6 Have students workin small groups to namethe direction of each vectorusing both the given angle and itscomplement.3. O yx7530 48 xO 37.5, –65.0 –307.3, –54.2 602.2, 668.8 Use compass directions to describe the direction of each vector.Example 2(page 453)4.5.NW15 WEEGPS Guided Problem Solving6.N20 NWL4EnrichmentEL2ReteachingL1Adapted Practice40 PracticeNameSSS40 east of south20 west of south15 south of westSketch a vector that has the given direction. 7–12. See margin.7. 508 south of east8. 208 north of westL39. 458 northeastClassL3DatePractice 8-6Perimeters and Areas of Similar FiguresFor each pair of similar figures, find the ratio of the perimeters and the ratioof the areas.1.82.43.354 cm35 cm4Find the similarity ratio of each pair of similar figures.4. two regular hexagons with areas 8 in.2 and 32 in.210. 708 west of north11. 458 southwest5. two squares with areas 81 cm2 and 25 cm212. 108 east of south6. two triangles with areas 10 ft2 and 360 ft27. two circles with areas 128p cm2 and 18p cm2Lesson 8-6 Vectors455For each pair of similar figures, the area of the smaller figure is given.Find the area of the larger figure.8.9.10.7 cmA 84 cm 2N10.N11.12 in.N12.5 in.A 20 in.27 in.5 in.A 18 in.215 cm8 in.For each pair of similar figures, find the ratio of the perimeters.11.70 12.13.A 50 cm2A 27 cm2A 4WEWEWE45 A 8 cm2A 1 in.2A 12 cm2in.214. The shorter sides of a rectangle are 6 ft. The shorter sides of a similarrectangle are 9 ft. The area of the smaller rectangle is 48 ft2. What isthe area of the larger rectangle?10 SSS455
Connection to History13. History Homing pigeons have the ability or instinct to find their way homewhen released hundreds of miles away from home. Homing pigeons carriednews of Olympic victories to various cities in ancient Greece. Suppose one suchpigeon took off from Athens and landed in Sparta, which is 73 mi west and64 mi south of Athens. Find the distance and direction of its flight.about 97 mi at 41 south of westFind the magnitude and direction of each vector. 14–16. See left.Example 3Exercise 13 Point out that our(page 453)modern Olympic Games, whichstarted in 1896, originated inancient Greece in 776 B.C.Alternative MethodExercise 14 Ask: If m represents14.14. about 707 mi;65 south of westmagnitude, what equationwould you write to find musing the Distance Formula?m "3002 1 6402 Whatequation would you write tofind m using the PythagoreanTheorem? m2 3002 6402Display the two equations, andhave students explain why theyare equivalent.15.N16.NN300 mi15. about 54 mi/h;22 north of eastWE20 mi/h1000 km640 mi16. 4805 km; 12 northof westWE50 mi/hWSSE4700 kmSIn Exercises 17–22, (a) write the resultant as an ordered pair and(b) draw the resultant. 17–22. See margin.Example 4(page 454)17.Tactile LearnersExercises 17–22 Have studentsuse pencils, straws, or otherstraight objects to model thevectors and their sums.18.yxO 6 4 24yy19.22 2 6 4 2 4x 2 OxO2 2Diversity20.Exercises 26, 27 Although thereare many mathematics problemsabout boats and currents, manystudents are unfamiliar with theidea of forces pushing in differentdirections. Help students relate theproblem to walking in a strongwind or swimming against acurrent.17. a. –9, –9 b.422.y 4 2x 2x1 222O y 4 2 O 4Write the sum of the two vectors as an ordered pair.Example 5(page 455)23. k2, 1l and k-3, 2l24. k0, 0l and k4, -6l25. k-1, 1l and k-1, 2l –2, 3 4, –6 –1, 3 Navigation The speed of a powerboat in still water is 35 mi/h. It is traveling on ariver that flows directly south at 8 mi/h.26. 35.9 mi/h; 12.9 south of west26. The boat heads directly west across the river. What are the resulting speedand direction of the boat? Round answers to the nearest tenth. See left.27. about 13.2 northof west27. At what angle should the boat head upriver in order to travel directly west?28. Aviation A twin-engine airplane has a speed of 300 mi/h in still air. Supposethis airplane heads directly south and encounters a 50 mi/h wind blowing dueeast. Find the resulting speed and direction of the plane. Round your answersto the nearest unit. 304 mi/h; 9 east of southy 3 O 9 3 918. a. –6, 2 b.4 619. a. –1, 0 b. 22 2yO x2yO x1BxApply Your Skills29. Critical Thinking Valerie described the direction of a vector as 358 south ofeast. Pablo described it as 558 east of south. Could the two be describing thesame vector? Explain. See left.29. Yes; both vectorshave the samedirection, butcould have diff.mag.45630. Error Analysis Ely says that the magnitude of vector k6, 1l is 3 times that ofvector k2, 1l since 6 is 3 times 2. Explain why Ely’s statement is incorrect. 6, 1 has mag. "37, but 2, 1 has mag. "5 .Chapter 8 Right Triangles and Trigonometry20. a. 1, –1 b.221. a. –8, 6 b.yy4xO2 245621.yx 2 OConnection to AlgebraExercise 31 Ask: What algebraicproperty does the ParallelogramRule establish? CommutativeProperty of Vector Addition2 8 4O x222. a. –2, –9 b.y 2 2O x 4 8
WvwuuvProblem Solving HintWWMath TipW31. The diagram at the left shows that you can addNGPS vectors in any order. That is, u v v u.100Notice also that the four vectors shown in red form30 a parallelogram. The resultant w is the diagonal ofW30 the parallelogram. This representation of vectoraddition is called The Parallelogram Rule. See margin.100a. Copy the diagram at the right. Draw a parallelogramSthat has the given vectors as adjacent sides.b. Find the magnitude and direction of the resultant. about 173 due eastExercise 33 After students writeWYou can also modelvector addition with theTriangle Rule as shownin Example 4, and byeither triangular halfof the diagram above.EExercise 35 Point out that thisexercise can be solved andanalyzed without drawinga diagram.W32. Use the diagrams below to write a definition of equal vectors.Exercise 45 Ask: Suppose ABdescribes walking due east at3 mi/h. What does BA describe?walking due west at 3 mi/hHave the class calculate thesum of AB and BA . 0WEqual vectors havethe same mag. anddirection.These vectors are equal.WExercise 49 Students may need33. Use the diagrams below to write a definition of parallel vectors.help extending the DistanceFormula to find the magnitudeof a vector in three dimensions. Ifpossible, provide a physical modelto help explain the formula.Vectors are n ifthey have thesame or opp.directions.WWVisual LearnersNo two of these vectors are equal.These vectors are parallel.their definitions, point out thatparallel vectors can have oppositedirections.No two of these vectors are parallel.34. Multiple Choice A Red Cross helicopter takes off and flies 75 km at 208 southof west. There, it drops off some relief supplies. It then flies 130 km at 208 westof north to pick up three medics. What is the helicopter’s distance from its pointof origin? C75 mi130 mi150 mi205 miWWWWW35. a. Find the sum of a and c , where a k45, -60l and c k-45, 60l. 0, 0 b. Writing Based on your answer to part (a), how can you describe a and c ?35b. a and c have mag. and opp.direction.WW36. Aviation In still air, the WP-3D (see below) flies at 374 mi/h. Suppose that aWP-3D flies due west and meets a hurricane wind blowing due south at 95 mi/h.What are the resultant speed and direction of the airplane to the nearest unit?about 386 mi/h at 14 south of westFlying into a HurricaneforWhen most pilots hear a “Time to fly.” Then again,anicOceonalNatidon’t work for then.and Atmospheric rops directly into hurricanmemcrewtThese aircraft carry eighofloadaand,tistsbers, up to ten scienofdata-collection equipment. Somethis equipment is in the WP-3D’salong “snout,” which also serves asthet,flighineroutalightning rod. IneWP-3D is struck by lightning threburnllsmagly,risinSurps.timefourorholes are the only damage fromthese strikes. To help overcome temninglightbyedcaussporary blindnespitflashes, the pilot sets the cocklights at the brightest level.Lesson 8-6 Vectors31. a.457N10030 30 WE100S457
4. Assess & ReteachGOPowerPointnlineHomework HelpVisit: PHSchool.comWeb Code: aue-0806Lesson QuizyProblem Solving HintNx1. Describe the vector as anordered pair. Round coordinates to the nearest tenth. 46.4, 18.7 2. Use compass directions todescribe the direction of ON .22 north of east3. Iris rode her bike 30 mi southand 16 mi west of her home.Her trip can be described bythe vector -16, -30 . Usedistance and direction todescribe the vector a secondway. 34 mi at about 28 westof south4. Write the vector v a b asan ordered pair.WW 2 1 1 2 3yO4x ba a b43. 0, –4 2 2byO4 xaI.1 2 3 4 5 6xb a bb yO 4x 2 a45. The vectors havethe same mag.; thevectors have opp.directions.46. Answers may vary.Sample: 7, 24 , –7, 24 , 7, –24 , 24, 7 47a. about 15 south ofwestb. about 6.7 hC458Alternative AssessmentHave each student write aparagraph explaining how thevectors -6, 8 and -8, 6 arealike and how they are different.83 11d 1 c dcd2124 25II.N39. c425 21d 1 c dcd0255WChallengeIII.NWNEWEESIIB. The plane cruises.SIC. The plane lands.41. Aviation The cruising speed of a Boeing 767 in still air is 530 mi/h. Supposethat a 767 is cruising directly east when it encounters an 80 mi/h wind blowing408 south of west. a. See back of book.b. 530, 0 ; –61.3, –51.4 a. Sketch the vectors for the velocities of the airplane and the wind.b. Express both vectors from part (a) in ordered pair notation.c. Find the sum of the vectors from part (b). 468.7, –51.4 d. Find the magnitude and direction of the vector from part (c).471.5 mi/h at 6.3 south of eastGive the sum of a and b . Show a and b and their sum in the coordinate plane.WWWWWWWWW42. a k-5, -2l, b k2, -5l43. a k5, -2l, b k-5, -2l42–44. See left.45. Writing How are vectors ABand BA alike? How are theydifferent? See left.W44. 3, –3 a 5, 2 5. An airplane has a speed of240 mi/h in still air. The planeheads due north andencounters a 30-mi/h windblowing due east. Find theresultant speed and direct
4. 5. 6. Sketch a vector that has the given direction. 7–12. See margin. 7. 508 south of east 8. 208 north of west 9. 458 northeast 10. 708 west of north 11. 458 southwest 12. 108 east of south W E N S 40 W E N S 20 W E 15 Example 2 (page 453) Oy x 30 75 O y 10 x 312 O x 48 900 Example 1 (page 452) 25 mi/h 7 mi/h x Quick Check 5 Q7 25R 7 25 .
I think of atomic vectors as “just the data” Atomic vectors are the building blocks for augmented vectors Augmented vectors Augmented vectors are atomic vectors with additional attributes attach
Two nonparallel vectors always define a plane, and the angle is the angle between the vectors measured in that plane. Note that if both a and b are unit vectors, then kakkbk 1, and ab cos . So, in general if you want to find the cosine of the angle between two vectors a and b, first compute the unit vectors aˆ and bˆ in the directions of a .
Draw vectors on your map from point to point along the trip through NYC in different colors. North Vectors-Red South Vectors-Blue East Vectors-Green West Vectors-Yellow Site Address Penn Station 33 rd St and 7th Ave Empire State Building 34th St and 5th Ave NY NY Library 41st and 5th Ave .
Chapter 6 139 Vectors and Scalars (ii) Vectors Addition is Associative: i.e. a b c a b c where . a , b . and . c . are any three vectors. (iii) O is the identity in vectors addition: Fig.9. For every vector . a O a Where . O. is the zero vector. Remarks: Non-parallel vectors are not added or subtracted by the .
6.1 An Introduction to Vectors, pp. 279-281 1. a.False. Two vectors with the same magnitude can have different directions, so they are not equal. b. True. Equal vectors have the same direction and the same magnitude. c. False. Equal or opposite vectors must be parallel and have the same magnitude. If two parallel vectors
Advanced Higher Notes (Unit 3) Vectors, Lines and Planes M Patel (April 2012) 1 St. Machar Academy Vectors, Lines and Planes Prerequisites: Adding, subtracting and scalar multiplying vectors; calculating angles between vectors. Maths Applications: Describing geometric transformations.
For two parallel vectors a b 0 4. The vector product of two vectors given in cartesian form We now consider how to find the vector product of two vectors when these vectors are given in cartesian form, for example as a 3i 2j 7k and b 5i 4j 3k where i, j and k are unit vectors in the directions of the x, y and z axes respectively.
vectors will approach 0, regardless of the vector magnitudes and . In the special case that the angle between the two vectors is exactly , the dot product of the two vectors will be 0 regardless of the magnitude of the vectors. In this case, the two vectors are said to be orthogonal.
