VECTORS

Teacher's Guide

This teacher's guide is designed for use with theVectors series of programs produced by TVOntario,the television service of The Ontario EducationalCommunications Authority. The series is available onvideotape to educational institutions and nonprofitorganizations. Vectors is the first series in the Con-cepts in Mathematics group.

OrderingIInformationTo order this publication or videotapes of theprograms in the series Vectors, or foradditional information, please contact one of thefollowing:

© Copyright 1988 by The Ontario EducationalCommunications Authority.All rights reserved.

Printed in Canada.

Teacher's Guide Writer : Ron CarrEditor: Kenneth DykemanDesigner: Roswita Busskamp

The SeriesProducer: David ChamberlainConsultants: Ron Carr and Bruce PetersProject Officer: John AmadioAnimation: Cinescan

CanadaTVOntario Sales and LicensingBox 200, Station QToronto, Ontario M4T 2T1(416) 484-2613

United StatesTVOntario U.S. Sales Office901 Kildaire Farm RoadBuilding ACary, North Carolina27511Phone: 800-331-9566Fax: 919-380-0961E-mail: ussales@tvo.or g

Videotapes BPN

Program 1: Follow That Arrow 280201Program 2: Finding the Resultant 280202Program 3: Ordered Pairs 280203Program 4: Resolving Without Grids 280204Program 5: Force 280205Program 6: Applying Forces 280206

CONTENTS

I ntroduction

1. Follow That Arrow

2. Finding the Resultant

3. Ordered Pairs

4. Resolving Without Grids

5. Force

6. Applying Forces

1

2

3

5

6

8

9

I NTRODUCTIONNearly all physical measurements may be divided intotwo major categories:• scalar quantities, which have only size or magni-

tude, and• vector quantities, which require two or more real

numbers for their specification. The measure-ments usually involved are size (magnitude) anddirection.Two interesting examples of vectors are velocity

and force. To describe the velocity of a pitchedbaseball or an airplane, one must state both speedand direction. A force is a vector quantity because thedirection in which it is acting and its strength are bothused to define it uniquely.

The six-program Vectors series examines theconcepts of velocity and force and shows how torepresent them as directed line segments. Notationfor vectors, their magnitudes, and their directions is

i ntroduced. Through the use of skills and relationshipsfrom geometry and trigonometry, the resultantof twovectors is found in right-angle and non-right-angleproblems. Using ordered pair notation, the resultantsof vectors on a grid are calculated with both algebraicand geometric methods. Practical applications involv-i ng vectors and operations with vectors are also intro-duced.

Vectors uses contemporary 3-D computer anima-tion and a humorous approach to enliven a branch ofmathematics with real-world applications. It is the firstseries in the Concepts in Mathematics group, and ispatterned on the highly successful Concepts inScience. Concepts in Mathematics will compriseminiseries on topics in trigonometry, algebra, statis-tics, and analytic geometry for senior highschool stu-dents.

FOLLOW THAT ARROW

ObjectivesAfter viewing the program and completing severalsuggested activities, students should be able to:1. give examples of and distinguish between scalar

and vector quantities;2. recognize the importance of defining both magni-

tude and direction of a vector;3. construct vectors as directed line segments,

using appropriate scales and angles;4. use correct notation to name vectors and express

their magnitudes and directions.

Before-Viewing Activitiesa) The program assumes that students are familiar

with the concept of scale. Review this conceptand draw some line segments to scale.

b) The program and the series assume that studentsare used to working with the metric system formeasuring lengths. Depending on the back-grounds of your students, spend some timereviewing this system.

c) Review the compass method of stating directioni n order that students may, when needed, indi-cate direction using notations such as North 52°East and 13° W of S.

PROGRAM 1 d) Ask students to name some everyday things thatare defined by two components. Examples areplaying cards (number and suit), and books (titleand author). This can introduce discussion of theconcepts of speed and velocity. Unlike speed,velocity requires two defining components -speed and direction.

Program DescriptionI n this first program, we are introduced to robot policeofficers Ed and Charlie, in their cruiser Chase One,who are attempting to capture a bandit. Guided bytheir Chief, flying overhead in a helicopter, they followi nstructions involving speed and direction as they tryto make a successful interception. They quicklyrealize the advantaaes of knowina about vectors.

After-Viewing Activitiesa) Students should practise drawing directed line

segments to scale to represent given velocities.

Examples:

= 50 km/h East,= 850 km/h Northwest,= 14 cm/s 30° E of S.

b) The class should practise naming vectors intriangle diagrams, using both lower-case andupper-case notations.

The concept of a vector and the need for twomeasurements to define it uniquely are presented.The velocity of 340 km/h East is used as an example.Directed line segments are used to represent vectors.Arrowheads indicate direction and a scale indicates

The program concludes with a return to the chasescenario. The concept of adding two vectors toproduce a third, called the resultant, is introduced.Determining the resultant vector will be discussed insubsequent programs.

c) The "bearing" system of direction could beintroduced as an alternative method of statingdirection. Note that every angle is measured fromNorth, moving in a clockwise direction.

PROGRAM 2FINDING THE RESULTANT

ObjectivesAfter viewing this program and completing severalreview and follow-up problems, students should beable to:1. understand that vectors can be added to produce

a third vector, called the resultant;2. realize that vectors can be considered to be in

any desired location in a plane, and can bemoved to new locations as long as the magnitudes and directions remain unchanged;

3. find the magnitude and direction of the resultantvectors given two vectors at right angles to eachother, or word problems involving velocities actingat 90° to each other.

Before-Viewing Activitiesa) This program, and the series, assumes that the

students can use calculators to do simpleoperations and find square roots. Practise someexamples.

b) Calculations in this series result in answers thatare correct to one decimal place. Review thisconcept and discuss other methods of expressingresults (significant digits, for example).

c) Spend some time using the Pythagorean Theo-rem to find the length of one side of a right-angledtriangle, given the lengths of the other two sides.

d) Review the definitions of the primary trigonometricratios using a right-angled triangle. Use theseratios to find lengths and angles in some tri-angles.

Program DescriptionThe program begins with the robot characters, Ed andCharlie, reviewing the content learned in Program 1.They see again that a vector may be representedusing a directed line segment where an arrowheadi ndicates direction and the length of the line segmenti s proportional to the magnitude of the vector. TheChief then sends Ed and Charlie on another missionand they decide to make use of the concept that thesum of two vectors produces a third vector called theresultant.

a) Students should construct reasonably accuratevector diagrams using appropriate scales andprotractors.

b) Before finding the resultant vector by calculations,the students should find the resultant by drawingaccurate right-angled triangles and measuring thel engths of the directed line segments and thesizes of the angles.

c) After finding resultant vectors from diagrams,students should find the resultant vector algebrai-cally, using the Pythagorean relationship for magnitude and the tangent ratio from trigonometry fordirection.

d) Problems involving velocity should be solved bystudents making working diagrams (not to scale)to illustrate the data. Use a calculator wherenecessary to find resultant vectors. Stress the useof correct notation. Remember that we are stillworking with only right-angled triangles.

Examples:1. Two vectors are defined as follows:

Draw a representative diagram and find the mag-nitude and direction of the resultant vector r ,

using Pythagoras and trigonometry.2. An airplane flying Northeast at 500 km/h encoun-

ters a crosswind blowing to the Southeast at80 km/h. Find the resulting ground velocity of theplane.

3. Repeat Example 1 but find

What law for vector addition does this illustrate?e) Having reviewed applications of the trigonometric

ratios before viewing the program, discuss withthe class a method for finding the magnitude ofthe resultant (hypotenuse) without using Pythago-ras; that is, by first finding an angle using thetangent ratio and then using the cosecant orsecant ratio to find the length. Compare themethods.

The angle (direction) of the resultant is foundusing the tangent ratio from trigonometry,

tan 8 = opposite sideadjacent side

in all calculations, the use of a calculator isstressed.

An example involving an airplane flying in acrosswind is presented. A representative diagram isconstructed and the Pythagorean relationship andtangent ratio are used to find the resulting groundspeed of the plane.

After-Viewing Activities

The notion of equal vectors is used to makevector diagrams to illustrate the resultant. (Thevectors in this program all "meet" at 90° angles.)

The magnitude of the resultant is found using thePythagorean Theorem,

PROGRAM 3ORDERED PAIRS

The problem of adding vectors that are notperpendicular is introduced. Vectors that are definedi n ordered pair notation are added to find the resul-tant.

ObjectivesAfter viewing this program and completing the sug-gested activities, students should be able to:1. draw vectors defined by ordered pairs on rectan-

gular grids;2. find the magnitudes and directions of vectors

defined by ordered pairs;3. understand the concept of resolving a vector into

its horizontal and vertical components and namethese components for vectors on a grid;

4. find the resultant of two vectors defined byordered pairs and, using components, calculatethe magnitude and direction of the resultant.

Before-Viewing Activitiesa) Review the Cartesian Co-ordinate System and

practise moving about the grid and plotting pointson it.

b) I ntroduce the concept of velocity on water meas-ured in knots (nautical miles per hour).

c) Review finding the magnitude and direction of theresultant of two perpendicular vectors.

d) I ntroduce the problem of finding the resultant oftwo vectors that do not form a 90° angle andobserve with the class that Pythagoras will notwork in this situation.

Program DescriptionThe opening sequence reviews the representation oftwo vectors by plane geometry and the skills neededto find the resultant (magnitude and direction of thesum) of two vectors acting at right angles.

A second way to describe vectors is introduced:vectors can be defined using ordered pairs andillustrated on a grid. The normal conventions todescribe movement to the right, left, up and down, areused and examples of vectors on a grid are shown.

Ed and Charlie experience a navigation problemthat introduces concepts surrounding the horizontaland vertical components of a vector and the additionof the ordered pairs representing the components.

A vector on a grid is presented and its orderedpair is found by considering its components. Theprocess of resolving a vector into its horizontal andvertical components is discussed.

(9,3) + (4,5) = (13,8)

The magnitude and direction of the resultant arecalculated using its components.

After-Viewing Activitiesa) Students should practise drawing directed line

segments on graph paper to represent vectorsdefined by ordered pairs.

Students should state the horizontal and verticalcomponents of each vector.

b) Students should find the resultants of adding"collinear" (parallel) vectors using grids.

c) Students should practise finding the resultant ofnon-perpendicular vectors, illustrating the vectorson a grid, and stating the components of theresultant. These components should be used tocalculate the magnitude and direction of theresultant using Pythagoras and the tangent ratio.

(4,0) + (3,0) = ?

(2,1) + (4,2) = ?

(-3,-2) + (-1,-2/3) = ?

(-4,2) + (2,-1) = ?

(1,2) + (4,1) _ ?

(3,3) + (-4,2) _ ?

(5,-2) + (-6,-3) _ ?

I nvestigate, using examples, to see if vectoraddition is associative.

e) What is the resultant vector when the vectors(3,4), (-5,1), (6,2), and (-4,-7) are added?What is the magnitude of the resultant? Thisvector is called the identity for vector addition.

PROGRAM 4RESOLVING WITHOUTGRIDS

ObjectivesAfter viewing this program, students should be ableto:1. find the resultant (sum) of three or more vectors

defined by ordered pairs;2. find the horizontal component and vertical compo-

nent of a vector that is not on a grid;3. find the magnitude and direction of the resultant

of two vectors that are not perpendicular byfinding its components.

Before-Viewing Activitiesa) Review the concepts involved with the horizontal

and vertical components of a vector on a grid.b) Review the use of Pythagoras and the tangent

ratio to find the magnitude and direction of the re-sultant of two perpendicular vectors.

c) Find the resultant, using a scale and protractor, oftwo vectors that are not perpendicular. For ex-ample, find the resultant of

Program DescriptionThis program opens with a review of the proceduresfor using a grid to find the resultant (sum) of two non-perpendicular vectors. The horizontal and verticalcomponents are used to calculate magnitude anddirection.

The concept of adding three or more vectorsdefined by ordered pairs is discussed. For example,

(4,4) + (3,0) + (0,-9) = (7,-5).

Skills and concepts involving vectors defined byordered pairs are now combined with those of vectorsdefined by directed line segments. The horizontalcomponent of a vector is calculated using the cosineratio and the vertical component is calculated usingthe sine ratio.

The problem of finding the resultant of two non-perpendicular geometric vectors is solved. Two

30 km/h N 25° W

50 km/h N 42° E

[(4,1) + (1,2)] + (-7,2)

(5,3) + (-7,2)

( -2,5)

vectors are shown and their magnitudes and angleswith the horizontal are given. The horizontal andvertical components of each vector are calculated andused to find the components of the resultants. Thesenumbers are then employed, using Pythagoras andthe tangent ratio, to find the magnitude and directionof the resultant.

Ed and Charlie re-appear briefly at the end of theprogram to introduce force.

After-Viewing Activitiesa) Students should practise adding three or more

vectors on a grid by adding the ordered pairs tofind the resultant, and then verify their work byactually drawing the given vectors and theresultant.

b) Find the horizontal and vertical components of thefollowing vectors.

4 4Find x + y.

400 km/h N 800 E

92 knots N 10° W

ii) A plane flying 610 km/h in the directionN 81° E encounters a wind blowing at 63 km/hfrom S 15° W. Find the resultant ground velocityof the plane.

d) Discuss other methods of finding the magnitudeand direction of the resultant of two non- perpen-dicular vectors. (The other methods involvemeasurement to scale, and the use of compo-nents.)Example:Given vectors

I ' Iy = 12.8 units

c) Solve the following problems.i) Find the resultant of the following pairs of

vectors by finding their horizontal and verticalcomponents.

.-I- PROGRAM 5FORCE

ObjectivesAfter viewing this program and participating in classdiscussions, students should be able to:1. understand the concept of force and realize that

force is a vector quantity because it has bothmagnitude and direction;

2. describe the forces acting on a body at rest andon an object that is falling;

3. understand the concept of equilibrium (staticcondition) with the sum of the force vectorsequalling zero;

4. resolve a force vector into its vertical and horizon-tal components to describe upward (downward)lift and horizontal push(pull) on an object.

Before-Viewing Activitiesa) Review the basic concepts: a vector has magni-

tude and direction; vectors can be added usingdirected line segments drawn to scale; vectorscan be resolved into components.

b) I ntroduce the concept of force through a discus-sion. Force is considered a vector quantitybecause it has magnitude and direction. Or,consider that force is equal to the product of massand acceleration,

Since acceleration is a vector, then force is also avector. (A scalar times a vector is a vector.)

Program DescriptionCharlie and Ed review methods to find the resultant oftwo non-perpendicular vectors. An example is workedusing horizontal and vertical components.

To this point in the series, velocity has been usedas an example of a vector quantity. Now a secondcommon and useful vector quantity, force, is introduced. Forces acting on objects that are at rest andobjects that are falling (gravity, reaction, tension) areexamined. The concept of equilibrium is discussedfor these objects. Directed line segments are used torepresent the forces involved, and vector diagramsare produced to illustrate the static condition.

The practical problem of pulling a sled is pre-sented and, using vectors and their horizontal andvertical components, an appropriate angle of pull is

found. I n another example, where a car stuck in snowi s being pushed, components are used again toillustrate the effective forward push and downwardpush at various angles. The newton is introduced asa unit of force.

After-Viewing Activitiesa) Discuss the forces acting on objects under the

following conditions. Have the students makevector diagrams (not to scale) to illustrate theseforces:• sitting at rest on a desk top• hanging from a single wire, and hanging from

two wires• falling and accelerating• falling at a constant velocity• standing at rest on an inclined plane• sliding on an inclined plane• being pulled by a rope

b) Borrow spring balances from the Physics Depart-ment in the school. Measure the forces for objectssuspended by one rope, by two or more ropes,and objects pulled in a variety of situations.

c) Discuss with the class the newton (N), the unit offorce commonly used in the metric system. Onenewton is the force that imparts to a one-kilogrammass an acceleration of one metre per secondper second (1 N _- 1 kg • m/s2). The unit offorce used in the British (Imperial) System is thepound (lb). For conversion purposes, 1 Ibf =4.448 N and 1 N = 0.2248 Ibf.

PROGRAM 6

ObjectivesAfter viewing this program and completing subse-quent suggested problems, students will be able to:1. resolve forces into vertical and horizontal compo-

nents;2. resolve forces into perpendicular and parallel

components;3. find the resultant of two or more forces using

components, Pythagoras, and trigonometry;4. solve force problems involving tension, lift and

push, gravity and inclined planes.

Before-Viewing Activitiesa) Review the concept of force as a vector quantity

and the resolution of a vector into its horizontaland vertical components.

b) Discuss familiar forces such as gravity, reactionforces, and friction.

Program DescriptionThis program consists of examples of forces ineveryday situations.

An automobile is moved along a roadway by twopeople pulling on two ropes making a 60° angle.Calculations are done to find the resultant of thesetwo forces to compare with the situation where the cari s pulled with a single rope. Horizontal and verticalcomponents, Pythagoras, and trigonometry are usedi n the calculations.

A second example involves the lifting and pushingof a wheelbarrow holding a heavy load of sand.Vertical lift and horizontal push are discussed andcalculated.

Two examples are presented related to ani nclined plane. First, the forces related to snow piledon a slanted roof are discussed. The force of gravityi s resolved into a component perpendicular to the roof(the load on the roof beams), and a componentparallel to the roof (the tendency to slide down). Theforce of friction is introduced and equilibrium isdiscussed. Finally, calculations with the classicsituation of a mass on an inclined plane complete theprogram and series.

APPLYING FORCES

After-Viewing Activities

g) An object is suspended from two wires that makean angle of 30°. Using spring balances, it isdetermined that the forces in the two wires (ten-sion) are 1.3 N and 0.8 N. Find the mass of theobject. [Hint: See question (e).]

d) Find the resultant of the two forces,

e)

f)

Find the equilibrant of the two forces in question(d). (The equilibrant is the force required to makethe system static, or exist in a state of equilibrium.Its magnitude is equal to the magnitude of theresultant, but it acts in the opposite direction.)An object is sitting on an inclined plane thatmakes an angle of 25° with the horizontal. Gravityis pulling the mass downward with a force of152 N. Find the components of this force that areperpendicular to the plane and parallel to theplane.

in the diagram.

b) Find the horizontal and vertical components of aforce of 28.3 N acting at a direction of 32° to thehorizontal.

c) A waterskier is being pulled by a high-poweredboat. The force along the tow rope is 300 N andthe angle between the rope and the water is 12°.What part of the force is propelling the waterskieralong the water?