Vectors
Student
Handout
Procedure
Didactical
Comments
Extensions
Vectors
Isabella Soletta, Italy
Characteristics of the activity
The module is divided into 5 independent
activities
Classes: 11-13
Time need: from 2 to 5 hours depending on
the activities chosen by the teacher
Prerequisites:
Students should know in advance
 the concept of vector
 Carnot theorem and sine formula
(only for in-depth examination of activity
1)
 Use of dynamometer (only for activity 5)
SIX plus THREE
gives always NINE?
Or might it give, e.g., SEVEN ?
With vectors this may happen,
let us learn how…
Learning objectives:
Students
 will exercise summing, subtracting and
decomposing vectors
 will use an application of vector decomposition by studying simulated forces on
incline
 will experimentally verify what they
studied within the simulation
Use of Technology (TI-NspireTM CAS):
 Using the dynamical geometry
 Using the spreadsheet
 Using the command ‘capture’
Tools, Materials, Documents
 TInspire software for PC or TInspire for handheld
 Only for activity 5: dynamometer (manual or digital), cart, scale, incline
Contents
Filename tns
Available in
http://pilotit.tinspire.ti.com/files/folders/
folder_f/default.aspx
Student handout
vect-INS.tns
Guided exercise
vect-ANA1.tns
Completed exervect-ANA2.tns
cises
Experimental data
Vect-DATA
In the section ‘Teaching suggestions’ the above documents are briefly summarized.
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Vectors
Student
Handout
Procedure
Didactical
Comments
Extensions
Viewscreens from TNS document
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Vectors
Student
Handout
Procedure
Didactical
Comments
Extensions
Vectors
Goals:





To become familiar with summation and subtraction of two vectors
To calculate the module and the angle of the sum vector
To exercise in vector decomposition
To study by simulation the forces acting on the incline while changing the inclination
To perform an experiment on the incline by measuring the inclination, the mass on
the incline, the net force (activity 6)
Materials:
Special material is needed only for activity 6. If a digital dynamometer (working both for pushing and pulling forces) is available, fairly accurate measurements will be possible. Also a
spring dynamometer may be used, but great care must be payed to the direction of the
spring. The mass sliding on the incline may be a lab cart.
Resources:
Experimental data in tns format.
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Student
Handout
Vectors
Procedure
Didactical
Comments
Extensions
The mathematical basis and the TI’nspire document construction
It is possible to define a series of operations on vectorial objects: here we will see how to
build them and how to work with them, e.g. by calculating:



A. The sum of two vectors
B. The difference between two vectors
C. The vector components
A. The sum of two vectors
Two vectors may be added ; the result is still a vector
c=a+b
Given two vectors applied to the same point, the sum vector may be graphically obtained as
the diagonal of the parallelogram having as sides the two vectors.
In the figure:
Thin arrows are the vectors to be summed
Bold dashed arrow is the sum vector
Thin dashed segments are the parallelogram sides used to draw the diagonal.
The sum vector module may be calculated using the formula:
c  a 2  b 2  2ab cos 
where the italic letters a,b, are the added vectors modules, c is the sum vector module,and α
is the angle between the two vectors to be added.
The sum vector direction may be calculated using the formula:
sin  
where
γ
α
b
c
b  sin 
c
i.e.
 b  sin  

 c 
  sin 1 
is the angle between the sum vector with the vector a
is the angle between the two added vectors
is the vector b module
is the sum vector module
The demonstrations, made using cosine and sine rules are shown in the section Extensions.
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Vectors
Student
Handout
Procedure
Didactical
Comments
Extensions
Subtraction between two vectors
The subtraction between two vectors may be graphically derived.
d=a-b
We draw the vectors a and b ; then we draw the vector –b, and we apply the parallelogram
rule to the vectors a and –b.
In the figure:
Thin arrows are the vectors to be subtracted
Bold dashed arrow is the difference vector
Thin dashed segments are the parallelogram sides used to draw the diagonal.
Activities 1 and 2 will help you understanding this topic
C. Vector decomposition
A vector may be decomposed into components along two directions. This is the inverse operation with respect to the vector summation: we look for two vectors whose sum is the given
vector.
Let us consider the most common case of decomposition into orthogonal components. We
draw the vector c and two orthogonal straight lines (r1 and r2) passing through the point
where the vector is applied. Then we draw the two lines parallel to r1 and to r2 , passing
through the end point of the applied vector c. The two lines intersect lines r1 and r2 in points
that mark the end points of the vectors c1 and c2 shown in the figure.
You may use activity 3 to experiment vector decomposition.
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Student
Handout
Vectors
Procedure
Didactical
Comments
Extensions
Forces on incline
Which forces do act on a body placed on inclined plane?
The body is subject to the gravitational force P (weight) and to plane reaction force R.
P= mg
where m is the mass and g the acceleration due to gravity
The gravitational force (P) is directed along the vertical and the plane reaction (R) direction is
perpendicular to the plane.
To calculate R the vector P is decomposed into a component F parallel to the plane, and a
component S perpendicular to the plane. The vectors R and S have equal module and opposite directions. The sum F of vectors P and R is the net force acting on the body: it is equal to
the weight P minus S (the weight component that is balanced by the plane reaction).
It may be shown that the module of the net force F is
F  mg 
h
l
where l is the incline length and h its height,
or, in terms of inclination angle :
F  mg  sin 
Activity 4 will help you to better understand this topic.
The laboratory activity suggested in act. 5 will enforce these concepts .
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Vectors
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Extensions
Procedures for the 5 activities
Most part of the instructions needed to perform the activities is contained in the document
ANA1. Hereafter only the essential steps are reported.
ACT 1: Sum and subtractions of two vectors (doc ANA1)
In G&G application, you can use two fixed module vectors applied to the same point. Change
the angle between the two vectors, and observe what happens to the sum vector. Work on
the difference vector: change the angle between two given vectors and observe what happens to the difference vector.
In calculator application you can verify the formulas used to calculate the module and the
angle of the sum vector.
Using G&G application you can build and verify demonstrations, as shown in the section Extensions.
2. Building the sum vector
Apply the parallelogram rule to build the sum and the difference of two given vectors.
3. Vector Components
Calculate the components of a vector in a cartesian reference frame. Find the components of
a vector along two given directions.
4. Forces on the incline
Investigate the forces acting on a body placed on an inclined plane, by decomposing the
weight vector.
5. Measure the force acting on a body placed on an inclined plane
Needed: cart, scale, dynamometer, plane, spacers.
Measure the cart mass. Calculate the weight.
Measure the plane length, use the spacers to change the incline height.
Measure the force acting on the cart placed on the incline and the incline height.
Repeat the force-heigth measurements at least 6 times and place the results in spreadsheet
Verify the relation
F  mg 
h
l
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Vectors
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Extensions
Classroom management tips:
These activities are independent: they may be performed completely or partially depending
on the class needs.

Activities 1 and 2 may be completed within one teaching hour.

Activity 3 needs not more than 1 teaching hour.

Activity 4 needs 1 teaching hour

Activity 5 needs at least 1 hour to perform the lab measurements plus 1 hour for data
analysis and making graphs.
The teacher, depending on the class skillness, may let the students working in small groups
by themselves (using-INS) or on solved exampled (using ANA1) that allow exploring the
proposed items. The last document (-ANA2), that includes the activities completely workedout ) is mainly aimed at the teacher who may use it as an example.
Some experimental data are also included, as described in the section PROCEDURE.
Documents description:
Vector-INS.tns: aimed at expert students who can work alone: includes only general instructions for performing the 5 activities.
Vector-ANA1.tns: the previous exercise partially worked-out: includes questions for the students who should answer and complete the exercise.
Vector-ANA2.tns: the previous exercise completely worked-out.
Vector-DATA.tns: experimental data.
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Vectors
Student
Handout
Procedure
Didactical
Comments
Extensions
Sum vector module and direction
The sum vector module may be calculated using the Carnot theorem (also named cosine
rule) as:.
c  a 2  b 2  2ab cos 
where the italic letters a,b, are the added vectors modules, c is the sum vector module,and α
is the angle between the two vectors to be added.
When the two vectors are orthogonal the formula expresses the Pitagora theorem.
Here is the demonstration
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Vectors
Student
Handout
Procedure
Didactical
Comments
Extensions
The sum vector direction may be calculated using the sine rule, expressed by the following
relation for the angles and the sides in a triangle:
sin  
where
γ
α
b
c
b  sin 
c
i.e.
 b  sin  

 c 
  sin 1 
is the angle between the sum vector with the vector a (DAB in the figure)
is the angle between the two added vectors (CAB in the figure)
is the vector b module
is the sum vector module
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