Course Profile   Geometry and Discrete Mathematics (MGA4U), Grade 12, University Preparation, Catholic

 

Unit 3:  Vector Applications

Time:  15 hours

 

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5

 

Unit Description

Cartesian vectors are represented in two-space and three-space as ordered pairs and triples, respectively. The addition, subtraction, and scalar multiplication of Cartesian vectors are investigated. Applications involving work and torque are used to introduce and lend context to the dot and cross products of Cartesian vectors. The vector and scalar projections of Cartesian vectors are written in terms of the dot product. The properties of vector products are investigated and proven. Many of the activities in this unit are more effectively completed using dynamic geometry software, but can be done with pencil and paper.

 

Activity 1:  The Sum of the Parts is the Same as the Whole

Time:  3.75 hours

Description

Cartesian vectors in two-space and three-space are introduced. Students use the right-handed coordinate system to represent Cartesian vectors in three-space. The properties of addition, subtraction, and scalar multiplication are applied to Cartesian vectors.

Strand(s) & Learning Expectations

Strand(s):  Geometry, Proof and Problem Solving

Overall Expectations

GEV.01 - perform operations with geometric and Cartesian vectors.

PSV.01 - prove properties of plane figures by deductive, algebraic, and vector methods;

PSV.02 - solve problems, using a variety of strategies.

Specific Expectations

GE1.01 - represent vectors as directed line segments;

GE1.06 - represent Cartesian vectors in two-space and in three-space as ordered pairs or ordered triples;

GE1.07 - perform the operations of addition, subtraction, scalar multiplication, dot product, and cross product on Cartesian vectors.

PS1.06 - demonstrate an understanding of the relationship between formal proof and the illustration of properties that is carried out by using dynamic geometry software;

PS2.01 - solve problems by effectively combining a variety of strategies;

PS2.02 - generate multiple solutions to the same problem;

PS2.03 - use technology effectively in making and testing conjectures.

Prior Knowledge & Skills

·         Represent vectors as directed line segments.

·         Name a vector using labels of points at the tail and head, as in .

·         Perform operations of addition, subtraction, and scalar multiplication on geometric vectors.

·         Determine the horizontal and vertical components of geometric vectors using the primary trigonometric ratios.

·         Perform basic graphing and construction functions using dynamic geometry software, including translations and dilations.

·         Prove properties of plane figures using deductive and analytic geometry and geometric vectors.

Planning Notes

·         Prepare worksheets and overhead transparencies.

·         Have a supply of overhead transparencies and pens available if student volunteers are going to present solutions to homework questions.

·         This activity is best carried out using The Geometer’s Sketchpad®. Students who have never had the opportunity to experiment with this tool quickly learn the skills required to carry out the investigations and are often able to suggest alternate procedures to improve them in very short order. The software provides the opportunity to form and test conjectures in less time than would be needed for a pencil-and-paper investigation. However, given that Cartesian vectors are a focus in this unit, many of the investigations can be modified and performed using graph paper. The teacher may wish to have part of the class investigate a property using dynamic geometry software and the other part investigate it using pencil and paper. Students may then compare processes and results and recognize that there are multiple ways to approach problem solving.

·         A utility for creating a ‘closed arrow’ segment may be available in The Geometer’s Sketchpad®. If so, it may be found in the script tools. It is useful for vector constructions that require the segment tool. Make this tool available on each computer. (Display, Preferences, More, Set Script Tool Directory, Continue, OK.) A new setting will appear on the tool bar under the Object Information icon. This is the Script Tool icon. (click on the Script Tool icon, Samples, Scripts, Utility, click on Closed Arrow.) If this is not available, simply direct students to use the segment tool.

·         Teacher access to a computer with an overhead projection unit and an overhead projector facilitates technical trouble-shooting, especially if the students have not previously worked with dynamic geometry software. Alternatively, the activities could be done as a demonstration with an overhead projector.

·         Samples of student work concerning the proof of geometric properties of plane figures completed in Units 1 and 2 may have been completed on chart paper or overhead transparencies. If so, these should be retrieved and used as a springboard for the development of proof of the same properties using Cartesian vectors on the third day of this activity.

Teaching/Learning Strategies

Student Activity

Students:

·         use dynamic geometry software to construct geometric and Cartesian vectors and investigate the properties of scalar multiplication, addition, and subtraction of Cartesian vectors in two-space;

·         express Cartesian vectors as ordered pairs and triples and in terms of the standard basis vectors, , , and, ;

·         solve problems involving applications of force and velocity using Cartesian vectors;

·         prove the properties of scalar multiplication, addition and subtraction of Cartesian vectors and extend their understanding to three-space;

·         prove the properties of plane figures using Cartesian vectors.

Teacher Facilitation

·         Begin by reviewing the addition of geometric vectors. Put the following problem on the overhead projector with a grid for sketching a diagram of the vector sum.

·         Three soccer players converge on the ball and kick it at the same time. The forces exerted should be considered to act concurrently and are: =15N acting N45°E;  = 20N acting E30°S;
= 25N acting W20°N. Represent the vector sum graphically and find the resultant force that acts on the ball. Ask students to suggest and discuss the drawbacks, e.g., determining angles between subsequent vectors and representing them geometrically, lengthy calculations using the sine and cosine laws, of this method.

·         Discuss the ease (or difficulty) with which this method could be used in three-space.

·         Direct students to work through the computer investigations. An overhead projection panel for the computer is useful if the students are not familiar with the software being used. Specific instructions for each step are listed in parentheses.

·         If time is limited, divide the investigations up among the students. Advise the students that they will be expected to present a summary of the results of one investigation.

·         Circulate during the investigations, providing prompts and technical advice where needed.

·         The goal of each investigation is stated up front.

Note: Students who are familiar with The Geometer’s Sketchpad® will not require the instructions that follow.

Student Worksheet: What’s Your Position?

As you complete these three investigations, pay attention to the orientation of the vectors that you construct. Use the naming convention that states the label of the point at the tail followed by the label of the point at the head, e.g., for , the point A is at the tail of the vector and point B is at the head.

Program Set-Up

·         Display, Label Options, Autoshow Labels for New Objects, OK.

·         Display, Preferences, set distance units to centimetres, angle units to degrees and precision for both to tenths, OK.

·         Graph, Create Axes, Show Grid, Snap to Grid.

·         Translate by rectangular vector. In some software applications, this setting may only be selected during the translation process.

·         If the closed arrow utility is used to construct a vector, there are four parts of the vector that must be selected in order to translate the entire construction. Hold down the shift key and select the point at the tail of the vector, the line segment, the point at the head of the vector and the triangle that forms the arrowhead.

·         It may be necessary during the investigation to use the labelling tool to change the point labels so they are consistent with the instructions. Select the Text Tool, double click on the label to get a text box for changing the label.

Part A: Position Yourself for Success (or Get the Point?)

The Ultimate Goal: Coordinate geometry provides us with a useful means of describing vectors. A vector that has been translated so that its tail is coincident with the origin is called a ‘position vector’. In this activity, the characteristics of position vectors will be investigated when you:

·         construct a vector on the Cartesian plane;

·         determine the horizontal and vertical components of the vector by constructing a horizontal line through the tail of the vector and a vertical line through its head. Pay attention to the directions of the components and their associated signs;

·         determine the relationship between the end points of the vector and the translation that will make the tail coincident with the origin and carry out the translation;

·         compare the coordinates of the point at the head with the horizontal and vertical components of the vector;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Click on the Script Tool icon and construct a vector from the point A(-6, 1) to the point B(1, 5). Label the points.

2.   Construct a horizontal line through A (select point A and the y-axis, Construct, Parallel Line) and a vertical line through B (select point B and the x-axis, Construct, Parallel Line). Construct the point at the intersection of these two lines (select the two constructed lines, Construct, Point at Intersection). Label it C.

3.   Determine the distances between points A and C and between points B and C. These distances are the magnitudes of the horizontal and vertical components respectively of vector . Hide the constructed lines at this time (select each line, Display, Hide Lines).

4.   What translation would be required to move point A to the origin? Apply the translation (Transform, Translate, By Rectangular Vector, enter the horizontal and vertical components).

5.   What are the new coordinates of the head, B¢ (select B¢, Measure, Coordinates)?

6.   Compare the coordinates of B¢ to the magnitudes of  and .

7.   How do the directions of vectors  and  compare to the signs of the coordinates of point B¢?

8.   How do the horizontal and vertical components of the translated vector  compare to the coordinates of point B¢ ? Comment on direction as well as magnitude. The vector  is referred to as a position vector.

9.   Given the vector  with points A(xA, yA) and B(xB, yB), suggest a formula to determine

a)   the translation that will move vector  to the origin;

b)   the horizontal and vertical components of the position vector of .

Test your conjectures by clicking on the head of the position vector and dragging it to locations in all four quadrants.

10.  A vector that has been translated so that its tail is at the origin of the Cartesian plane is called a position vector. Position vectors have interesting characteristics that make them convenient to work with. Summarize the results of your investigation of these vectors. Comment on any shortcomings of the dynamic geometry software when comparing the magnitudes of the components of position vectors to the coordinates of the points at the heads in quadrants 2 - 4.

Teacher Facilitation

·         Ask students to volunteer results, making sure that they arrive at the following conclusions: the horizontal and vertical components of  are the same as the coordinates of the point B¢; the components of the translation needed are the same magnitude as the coordinates of the point at A; but opposite sign; .

·         Question the class regarding the shortcomings of the process: the set-up does not allow for the negative components that occur in quadrants 2, 3 and 4 because the Distance measurement is being used.

Part B: It All Adds Up!

The Ultimate Goal: We will now move on to the addition of position vectors. In two-space, these vectors are represented as ordered pairs. Although these numbers look like the coordinates of a point, and happen to be the same as the coordinates of the head of the vector, they represent the magnitude and direction of the horizontal and vertical components of the position vector.

How do position vectors simplify the task of adding vectors? In this activity you will:

·         use the segment tool or closed arrow utility to construct three vectors, added head to tail. The tail of the first vector should be coincident with the origin;

·         construct the resultant vector that represents the sum;

·         compare the horizontal and vertical components of the original three vectors to those of the resultant vector and suggest a relationship among them;

·         test the relationship by changing the magnitudes and directions of the vectors. Save this construction for the next investigation;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Use the arrow (click on the Script Tool) or Segment Tool to construct three vectors that are added head to tail. Label the origin A, and then label the points that represent the ends of each vector B, C, and D.

2.   Construct the vector  that represents the vector sum (select points A and D, construct, segment). You may want to change its colour or set to thick line to set it apart from the others (select , Display, Line Style and/or Colour).

3.   Complete the first blank row of the table. Use the grid points on the graph to identify the horizontal and vertical components of vectors ,  and . Use these components to express each vector as a position vector. Pay particular attention to the direction and sign of each component.

Trial

Position vector

Position vector

Position vector

Sum of horizontal components of ,  and

Sum of vertical components of ,  and

Components of position vector

1

 

 

 

 

 

 

4.   Click and drag any of the points B, C, and/or D to change the vectors that are being added. Record the information for the new trial in the chart.

5.   Repeat Step 4 three more times. Leave the last sketch on the sketchpad for the next investigation.

6.   Summarize the results of this investigation.

Teacher Facilitation

·         Lead a brief class discussion to make sure that students have arrived at the conclusion that the sum of the individual horizontal and vertical components results in the horizontal and vertical components of the resultant vector

Part C: Can We Enlarge This Issue?

The Ultimate Goal: In this activity you will explore the properties of scalar multiplication of position vectors. Use the construction from the previous activity to:

·         dilate the vector sum about the origin;

·         observe the result and suggest a relationship between the dilation factor and the components of the vectors;

·         test the relationship by changing the dilation factor;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Click on the origin, A, and drag it to the lower left hand corner of the screen. Adjust the magnitudes and positions of , ,  and  so that they are confined to the first quadrant and that they do not occupy more than one quarter of the screen by clicking and dragging the points B, C, and D. This is necessary so that the dilated vectors will also be visible on the screen

2.   Mark the origin as the centre of the dilation (select A, Transform, Mark Centre “A”). Dilate the vector sum by a factor of two (Edit, Select All, Transform, Dilate, New Scale factor of 2). The image points of B, C and D should be labelled B¢, C¢ and D¢ respectively. Complete the first row of the table. Remember that you are working with position vectors, not points, and to watch the direction of each vector.

Trial dilation factor

Position vector

Position vector

Position vector

Position vector

Position vector

Position vector

Position vector

Position vector

2

 

 

 

 

 

 

 

 

3.   Undo (Edit, Undo) the dilation and repeat Step 2 for another dilation factor. Record the information for this dilation.

4.   Centre the origin on the screen. Repeat Step 2 once more with a dilation factor of !1.

5.   Summarize the results of this investigation. Discuss the direction of each position vector and its dilated image and compare their horizontal and vertical components. How is the sum of vectors affected by the dilation? How does a negative dilation factor affect vectors and their sum?

Teacher Facilitation

·         Lead a class discussion to check for understanding. Students should make the following observations: each original vector is parallel to its dilated image; the components of each image vector are found by multiplying the components of each original vector by the dilation factor; a negative dilation factor points the vector in the opposite direction.

·         Once students have had an opportunity to complete the investigations (it will probably require the entire first class allotted to this activity), assign each pair to summarize, with examples, the results of one investigation. The summaries will be presented on overhead acetates or on large sheets of chart paper the following day.

·         Presentations are made at the beginning of the next class. Monitor, and if needed, guide the summaries to general, algebraic statements of the properties of addition, subtraction, and scalar multiplication of Cartesian vectors. List statements on the board as they are presented.

·         Question the class about modifications to the properties and expression of Cartesian vectors in two-space that would be needed when working in three-space.

·         Introduce and define the right-handed coordinate system in three dimensions. Describe the coordinate planes and discuss how they divide three-dimensional space into octants. Express points as ordered triples, sketch the coordinate axes, plot points, and represent vectors as directed line segments and ordered triples.

·         It is useful to provide each student with three pieces of 25 cm by 25 cm wide meshed screen. Cut each piece from the midpoint of one side to the centre of the square. Cover the raw edges of screening with masking tape. These pieces of screening fit together to provide a model of the right-handed coordinate system in three dimensions. Pipe cleaners can be used to represent lines or vectors in three-space. These models come apart easily to be stored in zip-topped, plastic bags.

·         Develop the formula for determining the magnitude of vectors in two-space  and three-space . Determine  if  = (!2, 4, !3).

·         Define and develop the formula for determining the components of a unit vector in the direction of a given vector, expressed in Cartesian form (unit vector in the direction of ). Determine the components of a unit vector in the direction of  (!2, 4, 3).

·         Introduce the standard basis vectors , , and,  and demonstrate how they can be used to represent Cartesian vectors.

·         Revisit the ‘soccer ball’ problem. The given vectors can be resolved into components and manipulated as Cartesian vectors in order to reach the same solution.

·         Model the solution of problems involving the properties of addition, subtraction, and scalar multiplication of Cartesian vectors in two-and three-space. Include applications of force and velocity such as tension and projectile motion.

·         Assign questions that allow the students to consolidate their knowledge of the properties and manipulation of Cartesian vectors expressed as ordered pairs or triples or in terms of , , and,  and apply them to straightforward physics applications.

Note: It is often useful to ask for volunteers to commit to writing the solution to one of the assigned questions on an overhead transparency and presenting it the following day. The time required to go over trouble spots in the homework is greatly reduced when solutions are ready to go up on the overhead. This provides frequent opportunities for students to model exemplary solutions and to see how classmates are approaching the problem-solving process.

·         The third day of this activity should be devoted to proof of the properties of addition, subtraction, and scalar multiplication of Cartesian vectors and to their use in proving the properties of plane figures.

·         Students will have investigated and proven the properties of plane figures, i.e., the midpoints of the sides of a quadrilateral are the vertices of a parallelogram; the line segment joining the midpoints of two sides of a triangle is parallel to the third side, using deductive and analytic geometry (in Unit 1) and geometric vectors (in Unit 2). Review the structure required in the composition of a proof as was presented in these units, stressing the need for clarity, rigor, and adherence to form.

·         Revisit the proof of a property of a plane figure that was established in Unit 2. The property of triangles that states that the line segment joining the midpoints of two sides is parallel to the third side would work well here. Question the class and develop an outline of the successful strategies involving deductive and analytic geometry and geometric vectors that were employed. If samples of student work for this proof using alternate methods are available, use them as a starting point for this dialogue.

·         Develop, through class discussion, a strategy to establish the proof of this property using Cartesian vectors. Complete the proof as a class, modelling the desired structure. Emphasize the difference between the use of points in analytic geometry and that of Cartesian vectors. Reinforce the need for consistency between labels on the diagram and the vector notation used in the proof.

·         Introduce the use of Cartesian vectors as efficient tools for proving vector properties. Model a proof of one of the properties discussed on day two of this activity, e.g., Prove that  where k is any scalar and  and  are vectors in three-space.

·         Assign a set of proofs that allow the students to consolidate their understanding of the properties of Cartesian vectors and revisit the proofs of the geometric properties of plane figures that were established using other methods in the first two units.

Assessment & Evaluation of Student Achievement

Students should hand in completed investigations and summaries, where Communication and Thinking/Inquiry/Problem Solving can be assessed. Alternatively, verbal presentations could be assessed for Knowledge/Understanding, Thinking/Inquiry/Problem Solving, and Communication. The assigned set of proofs could be used to formatively assess Application and Communication, with emphasis on the correct use of mathematical conventions and symbols.

Accommodations

Student skill in using the dynamic geometry software may vary. It may be appropriate to pair students accordingly, as the focus of the investigation should be on the discovery of the mathematical principles. The size of the text and measurements on the computer screen can be increased to improve visibility (Display, Preferences, Text Styles).

Extension: The teacher may want to consider using the Texas instruments software, Derive (a trial version can be explored through the TI website). Among many other things, this software will perform vector algebra and, for the next unit, plot planes and show their intersections.

 

Activity 2: Work It Out!

Time:  2.5 hours

Description

The concept of physical work is used to introduce the dot product of geometric and Cartesian vectors.

Strand(s) & Learning Expectations

Strand(s):  Geometry, Proof and Problem Solving

Overall Expectations

GEV.01 - perform operations with geometric and Cartesian vectors;

PSV.02 - solve problems, using a variety of strategies.

Specific Expectations

GE1.01 - represent vectors as directed line segments;

GE1.02 - perform the operations of addition, subtraction, and scalar multiplication on geometric vectors;

GE1.03 - determine the components of a geometric vector and the projection of a geometric vector;

PS2.01 - solve problems by effectively combining a variety of strategies;

PS2.03 - use technology effectively in making and testing conjectures.

Prior Knowledge & Skills

·         Represent vectors as directed line segments.

·         Perform operations of addition, subtraction, and scalar multiplication on geometric vectors.

·         Determine the horizontal and vertical components of geometric vectors using the primary trigonometric ratios.

·         Determine vector and scalar projections of geometric vectors.

·         Define the change in position of an object as its displacement.

·         Perform basic graphing and construction functions using dynamic geometry software.

Planning Notes

·         Prepare worksheets and overhead transparencies.

·         Time constraints may prevent the teacher from covering both of the investigations in the first part of this activity. The ‘sugar bag’ investigation may be reduced to a 5-minute introduction to the concept of work or it could take the place of the first dynamic geometry investigation.

·         A 20 Newton spring scale, board protractor, inelastic string and a 2-kg mass (a 2-kg bag of sugar would work well) are needed. The spring scale should be available from the science department. Tie the string around the sugar bag as if wrapping a parcel. Experiment with the spring scale and protractor to ensure that a range of three force-angle combinations is possible and that the capacity of the scale is not exceeded. If graphing calculators are available, the spring scale can be replaced with a Calculator-based Laboratory (CBL) and a force probe.

·         Students should have access to computers with dynamic geometry software.

·         Access to a computer with an overhead computer projection unit or an overhead projector and data panel is helpful.

Teaching/Learning Strategies

Student Activity

Students:

·         investigate the role of applied force in determining the amount of work done on an object;

·         use dynamic geometry software to construct and calculate work done using geometric vectors;

·         define the dot product of geometric vectors based on observations made during the investigation.

Teacher Facilitation

·         Review vector and scalar projections and relate them to the context of work.

·         Introduce the activity and define work as the product of the component of the force in the direction of motion and the displacement through which the force acts.

·         Discuss and explain the fact that this product of two vectors results in a scalar quantity with units of joules (Newton-metres). It would be appropriate to explain, for the sake of those students who are not pursuing physics or chemistry at the senior levels, the use of joules as a common currency when expressing available energy and the capacity of a system to perform work. Discuss the effect of friction in overcoming inertia to get a sled moving, and the necessity to ensure that the sled is moving at a constant speed before the force measurement is taken

·         Put the following chart on the overhead for students to copy or provide photocopies.

Angle

Applied Force (Newtons)
Trial 1

Applied Force (Newtons)
Trial 2

Applied Force (Newtons)
Trial 3

Applied Force (Newtons)
Average

Horizontal Component of Average Force (Newtons)

Work Done (Joules)

 

 

 

 

 

 

 

·         This part of the investigation could be done as a demonstration or, if time and equipment availability permit, a small-group activity.

Note: This description will assume that a demonstration is used. Mark a distance of three metres on the floor. Discuss the equipment used and the basic procedure. Ask for three student volunteers. One will pull the ‘sled’, one will measure and monitor the angle, and the last will read the scale and record the observations on the overhead. Discuss the need to maintain a consistent force throughout the trials.

·         Direct the students to position the sled roughly 0.5 m in front the starting line and establish a string angle of approximately 30º. Pull the sled using a steady force and, once a constant speed is reached, take a force measurement. Maintain this applied force until the three-metre finish line is crossed. Repeat this step twice.

·         The previous step should be repeated for angles of 45º and 60º.

·         Once the data is collected, direct the students to calculate the average force for each trial.

·         Lead the students through the derivation of the expression to calculate the work done, i.e.,
Work = (Component of force in the direction of motion) (Displacement) .

·         Complete the chart with calculations of the components and the work done for each angle.

·         Discuss the relationship between the direction of the applied force and the amount of work done. Account for any unexpected results through the identification of sources of experimental error, e.g., variation in the applied force.

·         Define the result as the dot product and review characteristics that make it a scalar quantity.

·         Complete the following investigation using The Geometer’s Sketchpad®.

·         Set up the utility for creating a ‘closed arrow’ segment as in the previous activity. If it is not available, direct the students to use the segment tool.

Dynamic Geometry Investigation: It’s Off to Work We Go

Set-Up

·         Set distance units to centimetres, angle units to degrees and precision for both to tenths. (Display, Preferences)

·         In the graph menu, Create Axes, Show Grid, Snap to Grid.

·         Position the origin in the lower left-hand corner of the screen by clicking on the origin and dragging it. Do not alter the scales.

The Ultimate Goal: The link between the dot product of geometric vectors and the calculation of work is explored. You will:

·         construct two position vectors. One represents the applied force, and the other represents displacement, ;

·         construct a perpendicular from the head of the force vector to the displacement vector. Use the perpendicular to determine the projection of the force on the displacement, ;

·         have the dynamic geometry software calculate the work done in two ways:

Work; Work ;

·         compare the results of the two calculations and test the relationship by changing the magnitudes and directions of the vectors;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Construct two position vectors: one should be horizontal (to represent the displacement, ); the other should form an acute angle with the x-axis (to represent the applied force, ).

2.   Measure the magnitude of each vector (highlight the vector, Measure, Length) and the angle between them (highlight the head of one vector, the origin and then the head of the other vector, Measure, Angle). These quantities will be displayed on the screen. Record these measurements in the table in Step 6.

3.   Construct a perpendicular from the head of the force vector to the displacement vector (highlight the head of the force vector and the segment that represents the displacement vector, Construct, Perpendicular Line). To ensure that the line is perpendicular to the displacement vector and not the horizontal axis, you may want to drag the displacement vector off the x-axis before carrying out this construction.

4.   Construct a point at the intersection of the perpendicular and the displacement vector (highlight the perpendicular and the displacement vector, Construct, Point at Intersection).

5.   Measure and display the magnitude of the horizontal component of the Force vector, . This is the position vector that has its head at the point of intersection constructed in Step 4.

6.   Set up the following calculations on the dynamic geometry software: the product of the magnitude of the displacement vector and the magnitude of the horizontal component of the applied force ; the product of the magnitude of the displacement vector, the magnitude of the applied force and the cosine of the angle between them . Click on Measure and then Calculate. Highlight the desired measurement on the screen. It will appear on the calculator screen. Enter operations by clicking on the appropriate button on the calculator keypad. Click on Functions to access the cosine function. Record your observations in the table below.

Trial

1

 

 

 

 

7.   Change the position of the applied force and/or the displacement four more times. Record your observations and write a brief summary of the results. Compare the results of the calculations recorded in the last two columns and account for your observations. Include a definition of Work and relate it to the dot product of geometric vectors.

Teacher Facilitation

·         Lead the class in a discussion of the results. Summarize key information, i.e., the meaning of the dot product, the formula for the dot product of geometric vectors, the definition of work and the fact that it is an application of the dot product, on the board or overhead.

·         The scene is set for the following dynamic geometry software investigation by describing the scenario:

A sled is being pulled by a force  and moved through a displacement . We are going to investigate the use of Cartesian vectors to calculate the work done on an object and, by extension, the dot product of two position vectors.

Student Investigation: The Dot Product à la Carte(sian)!

Set-Up

·         Set distance units to centimetres, angle units to degrees, and precision for both to tenths.

·         Create Axes and Show Grid, Snap to Grid.

The Ultimate Goal: We have seen how Cartesian or position vectors simplify vector addition. How can position vectors be used to determine the dot product of two vectors? In this investigation you will suggest and test a formula for this vector product.

The Process

1.   Construct (5, 0) and s (10, 0). (Graph, Plot Points, Free Points). Label these points, F and s respectively (click on the Text Tool; double click on the label, edit the label using the text box).

2.   Create force and displacement position vectors by constructing segments that join points F and s to the origin (highlight the origin and point F, Construct, Segment; repeat with origin, point s).

3.   Measure and record the magnitude of the force vector, , the magnitude of the displacement vector,  (highlight the origin and the point at the head of the vector, Measure, Distance), and the angle q between them (highlight point F, the origin and then point s, Measure, Angle).

4.   Have the software calculate and display the Work done, namely . Click on Measure and then Calculate. Highlight the desired measurement on the screen. It will appear on the calculator screen. Enter operations as needed by clicking on the appropriate button on the calculator keypad. Click on Functions to access the cosine function.

Trial

q

1

(5, 0)

(10, 0)

 

 

 

 

2

(5, 1)

(10, 0)

 

 

 

 

3

(5, 2)

(10, 0)

 

 

 

 

 

 

 

 

 

 

4

(3, 0)

(6, 0)

 

 

 

 

5

(3, 2)

(6, 0)

 

 

 

 

 

 

 

 

 

 

6

(2, 0)

(8, 0)

 

 

 

 

7

(2, 4)

(8, 0)

 

 

 

 

 

 

 

 

 

 

 

5.   Repeat the process for each trial. Simply drag points F and s to each of the new locations.

6.   Conjecture a formula for the relationship between the components of the vectors and the amount of Work done. Record this conjecture.

Now, consider the scenario where an object is being moved through a vertical displacement.

7.   Repeat the first 5 steps for these vectors.

Trial

q

1

(0, 3)

(0, 8)

 

 

 

 

2

(4, 3)

(0, 8)

 

 

 

 

3

(9, 3)

(0, 8)

 

 

 

 

 

 

 

 

 

 

4

(2, 4)

(0, 9)

 

 

 

 

5

(5, 4)

(0, 9)

 

 

 

 

 

 

 

 

 

 

8.   Revisit the conjecture that was made in the first investigation. Is it still valid? If not, attempt to modify the formula.

9.   Once you think you have the formula, then try the following investigation (be sure to verify the result using

Trial

 (conjecture)

1

(5, 3)

(8, 1)

 

 

2

(2, 6)

(1, 4)

 

 

3

(-2, 1)

(-1, 6)

 

 

10.  Was your conjecture confirmed?

Teacher Facilitation

·         Have students present their conjectures on overhead transparencies.

·         Summarize the results on the blackboard and then lead the class to the formula for the dot product in terms of Cartesian vectors.

·         Assign problems work that allow the students to consolidate their knowledge and understanding of the dot product.

·         On day two, lead the class through the proof of the formula of the dot product of Cartesian vectors formally by relating it to the cosine law.

·         Model the solution to problems involving the relationship between the dot product of two vectors and the angle (acute and obtuse) between them. Focus on the use of Cartesian vectors to determine the geometric properties of plane figures. Include the determination of angles between a vector in three-space and each of the coordinate axes.

Assessment & Evaluation of Student Achievement

A formal, written report of the final investigation should be submitted and assessed for Thinking/Inquiry Problem Solving and Communication. Emphasis should be placed on the students’ ability to describe their process when carrying out the investigation and to draw conclusions using correct mathematical terms and vocabulary. Alternatively, assign a set of related problems and assess Knowledge/Understanding, Application, and Communication. Emphasize the need to communicate solutions with clarity, accuracy, and mathematical rigor.

 

Activity 3:  Dot’s Enough!

Time:  3.75 hours

Description

The properties of the dot product of Cartesian vectors are investigated. The dot product is applied to the determination of the geometric relationships between vectors, vector and scalar projections, and the geometric characteristics of plane figures.

Strand(s) & Learning Expectations

Strand(s):  Geometry, Proof and Problem Solving

Overall Expectations

GEV.01 - perform operations with geometric and Cartesian vectors;

PSV.01 - prove properties of plane figures by deductive, algebraic, and vector methods;

PSV.02 - solve problems, using a variety of strategies.

Specific Expectations

GE1.01 - represent vectors as directed line segments;

GE1.02 - perform the operations of addition, subtraction, and scalar multiplication on geometric vectors;

GE1.03 - determine the components of a geometric vector and the projection of a geometric vector;

GE1.06 - represent Cartesian vectors in two-space and in three-space as ordered pairs or ordered triples;

GE1.07 - perform the operations of addition, subtraction, scalar multiplication, dot product, and cross product on Cartesian vectors;

PS1.04 - prove some properties of plane figures, using vector methods;

PS1.06 - demonstrate an understanding of the relationship between formal proof and the illustration of properties that is carried out by using dynamic geometry software;

PS2.01 - solve problems by effectively combining a variety of strategies;

PS2.02 - generate multiple solutions to the same problem;

PS2.03 - use technology effectively in making and testing conjectures.

Prior Knowledge & Skills

·         Represent vectors as directed line segments.

·         Perform operations of addition, subtraction, scalar multiplication, and dot product on geometric and Cartesian vectors.

·         Prove properties of Cartesian vectors.

·         Determine vector and scalar projections of geometric vectors.

·         Perform basic graphing and construction functions using dynamic geometry software.

Planning Notes

·         Students work in pairs and require computers with The Geometer’s Sketchpad®.

·         The teacher should have a computer with an overhead projection unit.

·         Chart paper and markers are required.

·         Set up on the computer the utility for creating a ‘closed arrow’ segment as in the previous dynamic geometry activity. If it is not available, direct the students to use the segment tool.

Teaching/Learning Strategies

Student Activity

Students:

·         use dynamic geometry software to investigate properties of the dot product using Cartesian vectors and present their results;

·         prove properties of the dot product using Cartesian vectors in three-space;

·         work in pairs at first and then later, in larger groups (Days one and two);

·         use the dot product of Cartesian vectors to calculate scalar and vector projections.

Teacher Facilitation

·         There are many ways to present this material: as a teacher-led demonstration; as a dynamic geometry demonstration using an overhead projection unit; as groups assignments, each one being responsible for the investigation and presentation of one property; the investigations can be assigned as an independent study activity if computers are available outside of class time; or students can carry out all three investigations, working in pairs.

·         If the students carry out one or all of the three investigations, advise them that they will be called upon to present a proof of the results of one of them.

I’m Going Dotty!

Set Up

·         Graph, Create Axes, Show Grid, Snap to Grid

·         Display, Preferences, set distance units to centimetres, angle units to degrees and precision for both to tenths, OK.

·         Display, Label Options, turn off ‘Autoshow Labels for New Objects’, OK.

Investigation 1

When is ?

The Ultimate Goal: A dot product of zero is an important indicator of the geometric relationship between two vectors. In this investigation you will determine this relationship. You will:

·         construct two position vectors;

·         set up the calculation for the dot product ;

·         change the position and magnitudes of the vectors and make a sufficient number of observations to determine when the dot product is negative, zero, or positive. Save the construction for the next investigation;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Use the closed arrow to construct position vectors  = (3, 2), and  = (5, 1) (click on the Script Tool icon, begin at the origin, drag the head of the vector to the given point.)

2.   Click on the Text tool, label the vector by moving the cursor arrow to the middle of the vector, left click. Change the name of the vector to fit the investigation. Double click on the label, use the dialogue box that pops up to change the letter to a or b, as appropriate.

3.   Measure  and  (select a vector by clicking near its middle, Measure, Length).

4.   Measure the angle between  and  (holding down the shift key, select the head of , the origin and then the head of , Measure, Angle)

5.   Measure the cosine of the angle (Measure, Calculate, Function, cos, [click on the measurement of the angle]).

6.   Calculate the dot product  (Measure, Calculate, click on the measurement of the magnitude of , *, click on the measurement of the magnitude of ,*, click on the measurement of the cosine of the angle, OK)

7.   Hide the grid (Graph, Hide Grid) to facilitate reading the measurements and calculations on the screen. Make sure that the Snap to Grid option is still selected (a check mark is next to it).

8.   Click and drag the head of each vector, varying its length and the angles between vectors.

9.   Make conjectures about the following: When is ?; When is ?; When is ?.

10.  Once you have formed your conjectures, record supporting examples in the table below. Leave the last example on the screen. That construction will be used in the next investigation.

 

q

cosq

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11.  Summarize the results. Explain why the examples support your conjectures.

Teacher Facilitation

·         Circulate and ask leading questions when needed to guide students to conclude that:  when q is an acute angle;  when q = 90°;  when q is an obtuse angle.

·         Summarize the results on the board when students have completed the investigation.

Investigation 2

Verify that , for any scalar k.

The Ultimate Goal: In this activity you will investigate the effects of scalar multiplication on the dot product. You will:

·         use the construction from the previous investigation;

·         choose a value for scalar, m, and compare the dot products that result when m multiplies: the entire dot product;  only;  only;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Show the grid (Graph, Show Grid). Leave all measurements from the previous investigation on the screen.

2.   Apply a label of A to the head of . Apply a label of B to the head of vector . Hide the label at the origin (click on the Text Tool, double click on a label to change it, click on the feature labelled to hide it).

3.   Show the components of the position vectors  and  (select the point at the head of each vector, Measure, Coordinates). This will make it easier to construct and record the components of specific vectors.

4.   Construct  = (4, 2) and  = (6, 8) by clicking on the head of each vector and dragging it to the appropriate location.

5.   Record the value of the dot product  in the table in Step 14.

6.   Multiply (using a calculator if you want it displayed on the screen) the dot product by the scalar k, where k = 0.5. Record the result of this calculation in the table.

7.   Multiply  by 0.5, record the components of 0.5 in the table.

8.   Drag the head of  to the point that marks the head of 0.5.

9.   Record the value of the dot product in the column marked .

10.  Return  to its original position (Edit, Undo Translate Point).

11.  Repeat the dilation for  (Steps 7, 8).

12.  Record the value of the dot product in the column marked .

13.  Return  to its original position.

14.  Repeat Steps 1-13 with the values of ,  and k as shown in the table. Leave the constructions on the screen for the next investigation.

k

(4, 2)

(6, 8)

 

0.5

 

 

 

 

 

(-3, -1)

(5, -3)

 

2

 

 

 

 

 

(-6, 2)

(-2, -4)

 

-1.5

 

 

 

 

 

15.  Compare the values of ,  and  for each trial. Summarize, and account for, the results of this investigation.

Teacher Facilitation

·         Circulate and provide technical help and prompts when needed.

·         Lead a class discussion and put a summary of results on the board.

Investigation 3

Verify that

The Ultimate Goal: In this investigation, the distributive property of the dot product is verified.

To achieve this goal:

·         construct three position vectors ,  and;

·         calculate and display the dot products  and ;

·         construct  , calculate and display ;

·         compare the results of the calculations, suggest and test the relationship between them;

·         summarize the steps that were taken and describe the results of the investigation.

Technical Support

1.   Construct  = (2, !1) and  = (5, 3) by clicking on the head of each vector and dragging it to the appropriate location.

2.   On the computer screen, click on the calculation for  and drag it to a clear space on the right hand side of the screen.

3.   Construct  = (1, 6) (click on the Script Tool, move the cursor to the origin, click and hold down the left mouse button to drag the head of the vector to the point (1, 6); label the vector , and the end point C.

4.   Select point C, Measure, Coordinates.

5.   Measure the magnitude of  (select the line segment, Measure, Length) and the angle between  and  (highlight point A, the origin and then point C, Measure, Angle).

6.   Calculate  (Measure, Calculate, Length of , *, Length of , *, Function, cos, [angle measure], OK). Drag the calculation to a location immediately under the calculation of .

7.   Add  to . What translation will place the tail of  at the end of ? (select ; holding down the shift key, click on the origin, the line segment, the arrowhead and the point C, Transform, Translate, By Rectangular Vector, Horizontal Component, Vertical Component, OK).

8.   Construct  (select the origin and the point at the tip of the translated vector , Construct, Segment). Label the segment b + c.

9.   Measure the magnitude of  (select the segment, Measure, Length) and the angle between  and  (select the head of , the origin and then point A, Measure, Angle).

10.  Calculate  (As in Step 5). Drag this calculation to a location immediately under the calculation of  and complete the first row of the table below.

11.  Click on point A and/or point C and drag to a new location to test the relationship with other vectors four more times. Record the results in the table.

(2, !1)

(5, 3)

(1, 6)+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12.  What is the relationship between , , ?

Teacher Facilitation

·         Circulate and provide technical advice and prompts to ensure that the constructions yield the expected results.

·         Lead a class discussion and summarize the results of this investigation on the board.

·         Give each pair of students a piece of chart paper and a marker. Assign one of the following proofs to each pair of students, asking them to include a statement of the property, a supporting example in two-space, and a formal proof using Cartesian vectors in three-space. Proofs: Prove that if , then  for all , ; Prove that  for all , , ; Prove that = , for all , , .

·         Let each pair work on their proof until the end of the class and assign its completion and text book questions that allow the students to practise skills and consolidate understanding of the properties of the dot product for homework.

·         The next day, clear up any problems regarding the textbook work and then have the pairs of students responsible for each proof meet in one part of the room. They should compare their proofs and form a consensus regarding its presentation. A member of the group then copies the final draft of the exercise on a new piece of chart paper.

·         Circulate to each proof group to be sure they fully understand and can communicate the concept to the class.

·         Representatives from each larger group present their proof to the class.

·         Lead the class through the development of the formulas for Cartesian vectors:
scalar projection of  on =; vector projection of  on  =.

Model the solution to applications of these formulas.

·         Assign problems to consolidate knowledge and understanding and provide practice of related proofs (to address communication expectations) and applications. Advise the students that there will be a quiz the following day that covers the material studied in the unit so far.

·         On the third day of this activity, take up homework questions and then have the class write a quiz. A suggested format is provided here.

Quiz

1.   Given  = (2, 3, 7) and  (!3, 4, !1), find

a)  

b)  

c)   the angle between  and

d)   a unit vector in the direction of .

e)  

f)    a vector perpendicular to .

g)   the vector projection of  on .

2.   Sketch  (2, 3, 7) on a right-handed coordinate system in three-space.

3.   Given triangle ABC with vertices A(4, 1, 7), B(!2, 1, 1) and C(!3, 5, !6),

a)   determine the perimeter of ÎABC, correct to one decimal place;

b)   use vector methods to verify that ÎABC is a right angle triangle.

4.   A force  = (4, !7, 3), in Newtons, pulls a sled through a displacement  = (5, !6, !1) in metres. How much work is done on the sled by the force?

5.   Prove that  for all .

Extension: Have students research and write or download a program for their graphing calculators for the dot product. The program is brief, but would encourage the use of another, readily available technology. This program could be shared with the class and used as a tool when verifying that two vectors are perpendicular. This would provide a convenient check when students are learning the cross product.

Assessment & Evaluation of Student Achievement

Assess Communication during the presentation of proofs. The quiz can be used to assess Knowledge/Understanding (Questions 1 and 2), Application (Questions 3 and 4) and Communication (Question 5)

 

Activity 4:  Don’t Cross Me!

Time:  2.5 hours

Description

The cross product is introduced through a design scenario using specific vectors in three-space. The formula for the cross product is then generalized. Students are introduced to the moment of a force by considering the force required to close a door. They are shown that the magnitude of the cross product is the product of the distance from the turning point and the magnitude of the force perpendicular to that distance applied at that point. The right-hand rule is demonstrated as a means of determining the direction of the cross product. Properties of the cross product are introduced and applied to the solution of problems.

Strand(s) & Learning Expectations

Strand(s):  Geometry, Proof and Problem Solving

Overall Expectations

GEV.01 - perform operations with geometric and Cartesian vectors;

PSV.01 - prove properties of plane figures by deductive, algebraic, and vector methods;

PSV.02 - solve problems, using a variety of strategies.

Specific Expectations

GE1.01 - represent vectors as directed line segments;

GE1.02 - perform the operations of addition, subtraction, and scalar multiplication on geometric vectors;

GE1.03 - determine the components of a geometric vector and the projection of a geometric vector;

PS1.04 - prove some properties of plane figures, using vector methods;

PS2.01 - solve problems by effectively combining a variety of strategies;

PS2.02 - generate multiple solutions to the same problem.

Prior Knowledge & Skills

·         Represent vectors in geometric form.

·         Represent vectors in component form, including the use of the standard basis vectors , , .

·         Determine vector components.

·         Define the characteristics of a unit vector.

·         Use the dot product to determine the components of a perpendicular vector.

Planning Notes

·         If desired, prepare a wooden brace to demonstrate the footbridge application for the introduction of this activity. This need not be elaborate: two planks (1" by 4") could be nailed together to form a cross. A third, short length of lumber (2" by 2") would be used to show the orientation of the supporting beam.

·         Demonstration of the right hand rule requires a wrench, a large bolt, and a piece of wood with a pre-drilled hole in which to turn the bolt.

·         Demonstration of torque requires a force spring scale, metre stick, masking tape, protractor, and a door of uniform thickness. Experiment with the spring scale to ensure that a range of three-force distance combinations is possible and that the scale measure the appropriate range of forces for this demonstration. Prepare an overhead of the procedure and the table needed for this demonstration and have a marker available to record data on the table. This investigation could also be carried out by the whole-class, divided into pairs. However, time constraints and the availability of a sufficient number of doors may make this impractical. A CBL and force probe can take the place of the spring scale.

Teaching/Learning Strategies

Students:

·         derive the formula for the components of the cross product of two vectors in three-space.

Teacher Facilitation

·         Put the following problem up on the overhead or the board: A wooden footbridge is to be constructed. The directions of two intersecting beams can be described with the vectors  = (4, 7, 1) and
 = (
!3, !4, !2). A third beam, perpendicular to the first two, is to be attached at the point of intersection. Describe the direction of this third beam with a Cartesian vector. A simple model may be constructed with scrap lumber to help students visualize the application.

·         Lead a class discussion to the conclusion that the third vector,  = (x, y, z) will have a dot product of zero with each of vectors  and , since  is perpendicular to  and .

·         Instruct the students to determine the possible components of . This may be done individually or in pairs.

·         Circulate, providing prompts when needed. As students have not formally encountered systems of equations in three unknowns or parametric solutions, they will require support. Monitor their progress and, if necessary, return to a whole group discussion of the obstacles presented by this problem. If students have found components for vector , record them on the board and compare the solutions.

·         Question the class regarding the common features of the solutions. They should be lead to the conclusions that: the vectors are all parallel; they may be in opposite directions; their magnitudes need not be the same, and each yields a dot product of zero when paired with either of the original vectors,  and .

·         Generalize this relationship using vectors  and  = (x, y, z), leading to the result

·         Show the class how to set up and use the following mnemonic device as way to remember the cross product relationship.

 

 

·         Demonstrate the right-hand rule using a screwdriver and screw. Ask for a volunteer to screw the screw into a piece of wood that has been pre-drilled. Instruct the class to curl the fingers of their right-hands in the direction of rotation. To screw the screw into the wood their fingers curl in a clockwise direction and their thumbs point down. Note that the screw is also moving down into the wood. To remove the screw from the wood, the screwdriver is moved in a counter clockwise direction. Using the right-hand rule results in a ‘thumbs up’, which corresponds to the upwards direction of the screw.

·         Lead a class discussion regarding the implications of the right-hand rule and confirm the non-commutative nature of the cross product. Write any two vectors on the board (i.e.,  = (2, 4, !3) and  = (!5, 3, 2)). Ask for two volunteers: the first one calculates ; the second one calculates . Monitor the board work to ensure the correct outcome of this exercise.

·         Introduce the characteristics of scalar multiplication of the cross product, , and verify the property with any two vectors in three-space.

·         Assign the formal proofs of the non-commutative and scalar multiplication properties of the cross product. Ask for volunteers to write one of the proofs on an overhead transparency. If possible, assign each proof to two or three students to ensure adequate material for discussion the following day. Give two acetates and an overhead marker to each volunteer. Also assign problems to consolidate knowledge and understanding of the cross product.

·         The next day, have the volunteers present their proofs. Monitor the presentations and ask the class to discuss the merits (i.e., form, approach) and any needed modifications at the end of each. Guide the discussions to ensure that students have included all required elements of each proof.

·         Introduce the Distributive Law of the cross product,  and verify the property with any three vectors in three-space. This property can be used to derive the formula for the components of the cross product using the standard basis vectors, as follows:

·         Recall that  = (1, 0, 0),  = (0, 1, 0) and  = (0, 0, 1). These vectors have a magnitude of 1 and are perpendicular to each other.

·         Lead a discussion about the results of finding the cross product of two of these vectors. For example, applying the formula for the cross product leads to  and that . Now repeat for . In this case, the result is the zero vector. Use this result to establish the property of any two parallel vectors  and , namely, . Confirm this property later using the formula for the magnitude of the cross product.

·         Now, consider the case of two vectors  = (a, b, c) and  = (d, e, f). These vectors can be written as  =  and  = . Thus,

  =

Simplify using the cross product

   =

Factor

   =

This expression can be compared to the one derived earlier in the activity.

Student Activity

Students:

·         carry out a demonstration of torque in order to build an understanding of the meaning and characteristics of the cross product and to develop the formula for the magnitude of the cross product, .

Teacher Facilitation

·         Prepare the three loops on the door in advance. Using several pieces of masking tape, tape a small loop of string to the door near the edge of the door away from the hinge. Repeat for the middle of the door (at the same height) and at a point near the hinge.

·         Remind the students that all previous applications of force involved concurrent forces, and that any resulting motion could be described as a translation.

·         Introduce torque as a twisting effect caused when a force, , is applied to an object at some point with a position, , relative to the centre of rotation. The torque is dependent on the amount of force, where it is applied, and the angle between the line of action of the force and the object and is calculated by finding the cross product . The activity below demonstrates that, as the distance from the turning point increases, the perpendicular force required to close the door decreases.

·         Put the prepared transparency on the overhead and outline the procedure.

·         Ask for three volunteers: one to record the data on the overhead transparency; one to measure the distances and then hold the protractor in order to ensure that the force is perpendicular to the door; and the third to pull the door closed and read the force measurement off the scale.

Demonstration Activity:  Close the Door!

1.   Measure the distance from the hinge to each loop and record it in the table.

2.   With the door open, hook the spring scale to a loop and slowly pull on it to close the door, observing the reading on the scale at the instant the door begins to move. Be sure that the spring scale is perpendicular to the door at all times. Repeat several times for each loop and calculate the mean force required to move the door. Record your observations in the table below.

3.   Calculate the product of the distance from the hinge () and the force () required.

 

Near Hinge

Middle

Close to edge

Distance (m)

 

 

 

Force (N)

 

 

 

(Distance) (Force)

 

 

 

Teacher Facilitation

·         Compare the magnitudes of the forces required in the three cases. Discuss the result that a smaller force applied farther from the centre of rotation is just as effective as a larger one applied close to the centre of rotation. Ask the class to suggest applications of this knowledge: How can we increase torque, e.g., large pipe wrenches, steering wheels? How can we decrease torque, e.g., lowering a centre of gravity to prevent tipping in vehicle design, when skiing?

·         With this activity the teacher can introduce the notion that the magnitude of the torque is determined by finding the product of the magnitude of the component of the force perpendicular to door times the distance from the hinge. The distance is the magnitude of the vector, , that describes the position of the point of application of the force relative to the centre of rotation. This leads to the formula .

·         Use the right-hand rule to discuss the direction of the torque.

·         Model the solutions to the following problems:

1.   A 15 N force is applied to the end of a 35 cm wrench at an angle of 70°. What is the magnitude of the component of the force perpendicular to the wrench? Calculate the magnitude of the torque of this force about the other end of the wrench. What force would have to be applied at the same point at an angle of 45° if the same torque is required?

2.   The position of the end a support beam for a light fixture can be described with the vector
 = (1.5, 2, 0), in metres. A supporting force,  = (
!3, !4, 8), in Newtons, is applied to the end of the beam. Calculate the torque about the other end of the beam.

·         The teacher should provide a sketch of the vectors in Question 2 on the right-handed coordinate system to illustrate how the torque relates to  and . Show how the right hand rule confirms the result.

·         Assign related problems to consolidate skills, to provide an opportunity to practise their applications, and to engage students in rich, multi-faceted tasks.

Extension: Students can research and write or download a program for the graphing calculator that determines the cross product (visit the Texas Instruments website). Once again, this could be shared with the class and used in this unit and in Unit 4, (when writing Cartesian equations of planes).

Assessment & Evaluation of Student Achievement

Students hand in a report that summarizes the results of the torque investigation, with emphasis paid to the assessment of Communication (through the use of mathematical symbols and vocabulary) and Application (through a discussion of the factors affecting the increase or decrease of torque).

 

Activity 5: Summative Assessment

Time:  2.5 hours

Description

This assessment includes questions that test the students’ acquisition of the expectations in this unit across all of the categories of the Achievement Chart. The tasks can be given as part of a paper-and-pencil test, individual or group investigations, an assignment or some combination of these.

Strand(s) & Learning Expectations

Strand(s):  Geometry, Proof and Problem Solving

Overall Expectations

GEV.01 - perform operations with geometric and Cartesian vectors;

PSV.01 - prove properties of plane figures by deductive, algebraic, and vector methods;

PSV.02 - solve problems, using a variety of strategies.

Specific Expectations

GE1.01 - represent vectors as directed line segments;

GE1.02 - perform the operations of addition, subtraction, and scalar multiplication on geometric vectors;

GE1.03 - determine the components of a geometric vector and the projection of a geometric vector;

GE1.06 - represent Cartesian vectors in two-space and in three-space as ordered pairs or ordered triples;

GE1.07 - perform the operations of addition, subtraction, scalar multiplication, dot product, and cross product on Cartesian vectors;

PS1.04 - prove some properties of plane figures, using vector methods;

PS1.06 - demonstrate an understanding of the relationship between formal proof and the illustration of properties that is carried out by using dynamic geometry software;

PS2.01 - solve problems by effectively combining a variety of strategies;

PS2.02 - generate multiple solutions to the same problem;

PS2.03 - use technology effectively in making and testing conjectures.

Prior Knowledge & Skills

Students should possess a comprehensive knowledge of the concepts introduced and extended throughout this unit.

Planning Notes

·         Computers could be made available to the students if dynamic geometry software is needed for forming and testing conjectures or solutions.

Teaching/Learning Strategies

Teacher Facilitation

·         Each task in this assessment is preceded by a suggestion of the skill categories that would be most applicable to the given task.

·         This assessment could assume several forms: it could be used as a unit test, to be completed individually by the students; pairs or groups of students may perform an inquiry prior to the assessment period. Each student then uses the results of the inquiry to prepare an individual response on the ‘test day’. The tasks in Part B would be well suited to this format.

·         It is recommended that this summative assessment take place over two days to allow for a thorough evaluation of student performance.

·         Each task in this assessment is preceded by the skill categories that would be most applicable to the given task ([K/U] indicates Knowledge/Understanding, [T/I/PS] indicates Thinking/Inquiry/Problem Solving, [C] indicates Communication, and [A] indicates Application).

Student Activity

Part A

1.   [K/U, A] The displacements of two ships, A and B, two hours after leaving from the same port can be represented with position vectors  (20, 50, 0) and  (!60, !10, 0). Assume that the port is located at the origin and that all units are in kilometres.

a)   How far from the port is each ship?

b)   How far apart are the two ships?

c)   The displacement of a bird from the port can be described with the vector –65 – 8+0.5.

i)    How high above the water is the bird?

ii)   How far from ship B is the bird?

d)   What will be the position vector of the displacement of ship A from the port 3.5 hours after leaving the port? Assume that the direction and speed of the ship are constant.

2.   One point on the surface of a sphere centred at the origin has the coordinates (8, 3, 5)

a)   [K/U] Represent the radius that connects this point to the origin as a position vector.

b)   [K/U, A] Determine the angle that this radius makes with each of the coordinate axes.

c)   [K/U] Suggest the position vector of a radius of this sphere that lies below the xy-plane.

3.   [K/U] If and  find:

a)  

b)   a unit vector in the direction of

c)  

d)  

4.   [A] How much work is done by a force, in Newtons, represented by the vector  = (!4, 7, 1) if it results in a displacement, in metres, represented by the vector  = (!5, 3, 8).

5.   [A] If vectors  and  are perpendicular and