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.
·