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
a
·
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
so
=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 Su
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 o
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 |
Sum of vertical components of |
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 o
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 ‘so
·
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 su
·
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.
Acc ommodations
Student
skill in using the dynamic geometry software may vary. It may be appropriate to
pair students a
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 a
·
A
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) |
Applied Force (Newtons) |
Applied Force (Newtons) |
Applied Force (Newtons) |
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. A
·
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 a
|
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 a
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 a
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1 |
(5, 0) |
(10, 0) |
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2 |
(5, 1) |
(10, 0) |
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3 |
(5, 2) |
(10, 0) |
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… |
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4 |
(3, 0) |
(6, 0) |
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5 |
(3, 2) |
(6, 0) |
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… |
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6 |
(2, 0) |
(8, 0) |
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7 |
(2, 4) |
(8, 0) |
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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.
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q |
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1 |
(0, 3) |
(0, 8) |
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2 |
(4, 3) |
(0, 8) |
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3 |
(9, 3) |
(0, 8) |
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… |
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4 |
(2, 4) |
(0, 9) |
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5 |
(5, 4) |
(0, 9) |
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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 
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1 |
(5, 3) |
(8, 1) |
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2 |
(2, 6) |
(1, 4) |
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3 |
(-2, 1) |
(-1, 6) |
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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.
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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.
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(4, 2) |
(6, 8) |
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0.5 |
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(-3,
-1) |
(5, -3) |
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(-6, 2) |
(-2,
-4) |
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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.
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(2, !1) |
(5, 3) |
(1, 6)+ |
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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 