Senior High School
GENERAL PHYSICS 1
Quarter 1 – Module 2
VECTORS AND VECTOR ADDITION
Department of Education ● Republic of the Philippines
GENERAL PHYSICS 1
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Quarter 1 - Module 1: VECTORS AND SCALAR FORCES
First Edition, 2020
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Senior
High
School
Senior
High
School
GENERAL
PHYSICS 1
Quarter 1 - Module 1
VECTORS AND VECTOR ADDITION
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Table of Contents
How to learn from this Module ................................................................................................................ i
Icons of this Module ................................................................................................................................... ii
What I Know ................................................................................................................................................iii
Lesson 1: VECTORS AND SCALAR FORCES
What I Need to Know..................................................................................................... 1
What’s In …………………………………………………………………………1
What’s New: (Activity 1 Graphical representation of vectors) ......................... 2
What Is It ........................................................................................................................... 2
What Is It (Discussion) .................................................................................................. 4
What’s More: (Activities) ............................................................................................. 4
Lesson 2: RESULTANT OF VECTORS
What’s In............................................................................................................................ 4
What I Need to Know..................................................................................................... 4
What Is It ( Discussion) .............................................................................................. ..5
What’s More: (Example: Social Media Scenario) .............................................. .9
What I Have Learned..................................................................................................... 10
Assessment: (Post-Test)………………………………………………………………………..11
Key to Answers......................................................................................................................................... 12
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
•
Take your time reading the lessons carefully.
•
Follow the directions and/or instructions in the activities and exercises diligently.
•
Answer all the given tests and exercises.
Icons of this Module
What I Need to
Know
This part contains learning objectives that
are set for you to learn as you go along the
module.
What I know
This is an assessment as to your level of
knowledge to the subject matter at hand,
meant specifically to gauge prior related
knowledge
This part connects previous lesson with that
of the current one.
What’s In
What’s New
An introduction of the new lesson through
various activities, before it will be presented
to you
What is It
These are discussions of the activities as a
way to deepen your discovery and understanding of the concept.
What’s More
These are follow-up activities that are intended for you to practice further in order to
master the competencies.
What I Have
Learned
Activities designed to process what you
have learned from the lesson
What I can do
These are tasks that are designed to showcase your skills and knowledge gained, and
applied into real-life concerns and situations.
II
What I Know
(Pretest)
Multiple Choice. Select the letter of the best answer from among the given choices.
Direction: Multiple choices. Select the letter that correspond to the best answer.
1. Which of the following statement is true in determining the resultant of two or more
vectors acting together in the same direction?
a. Add the given vectors and take the common direction
b. Subtract the given vectors and take the common direction
c. Get the product of the given vectors and take the two direction
d. Add the given vectors and take the two direction
2. All statement are true about vectors, except;
a. Vectors are quantities with specified magnitude and direction
b. Vectors are quantities that have magnitude only
c. Vectors of two or more can be add by component method
d. Vectors can be add and take the common direction
3. A quantity that can be completely described by a single value called magnitude.
a. Vector
b. magnitude
c. scalar
d. velocity
4. The following statement is an example of scalar, except;
a. An ant crawl on top of the table 15 cm.
b. The troop of soldier walking 20km to northward direction
c. A bunch of flowers weighing 5 kilograms
d. A car runs fast 50km/h
5. A quantity that have magnitude and direction.
a. Vector
b. scalar
c. displacement
d. velocity
6. What is the vector some of 3 unit east and 5 unit east?
a. 5 units east
b. 6 units east
c. 7 units east
d. 8 units east
7. To find the resultant of two vectors with opposite direction, you need to;
a. Get the difference and take the direction of the vector with the greater value
b. Get the sum and take the direction of the vector with the greater value.
c. Get the product and take the direction of the vector with the greater value
d. Get the product and take the direction of the vector with the smaller value
8. Methods which are used to determine the vector sum or resultant of two or more
vectors, except;
a. Parallelogram
b. polygon
c. component
d. symmetrical
9. Which of the following is true about graphical method in calculating resultant vector?
a. The vectors to be added are drawn according to a convenient scale
b. The vectors to be subtracted are drawn according to a convenient scale
c. The product of vectors are drawn according to a convenient scale
d. The vectors to be subtracted are directly drawn.
10. Since vectors are quantities with specified magnitudes and direction, the most
appropriate representation of the vector is;
a. Arrow
b. line
c. curve
d. angle theta
11. The formula used in obtaining the magnitude of the resultant in component method.
a. Pythagorean theorem b. Polygon method c. arc tangent
d. Trigonometric
12. Which of the following is an example of vector quantity?
a. 25km/h
b. jumping 20 seconds
c. 50 grams
d. 5cm, east
13. What is the difference of the two vectors 8 units, east and 5 units, west?
a. 3 units, west
b. 3 units, east
c. 4units west d. 4units, east
14. What is the exact sum of vectors A= 10cm, east and B= 5cm, east?
a. 5 cm, east
b. 5 cm, west
c. 15cm, east
d.15cm, west
15. The net displacement obtained from two or more vectors.
a. Scalar
b. resultant
c. sum
d. force
Lesson
1
VECTORS AND SCALAR
FORCES
Physical quantities are all around us, the number of hours we spend in school or in
our work, the speed and direction of the jeepney that we ride on everyday, and the amount
of food that we buy. How to describe accurately these physical quantities that involved?
Physical quantities are divided into two groups, the first group consists of length,
area, volume, and speed while the second group consists of force, acceleration, and
velocity. Which group do you think gives a clearer picture of quantities? When we describe
the speed of the wind, we say it 69 kmh; but when we talk about its velocity, we express it as
69 kmh, North of West of Manila. Could you find any differences between these two
quantities? Which gives an accurate and precise description of the wind? Why?
This module will present information regarding another physical quantity, force. This
quantity is common thing to us. Pulling a table, lifting heavy baggage, moving a chair, or
opening a window involves force. Pushing a table requires a greater exertion of the muscles
than pushing a chair. Why is this so? Why is it easier to transport a block of wood on a cart
with wheels than to carry it on one’s shoulder?
In this module you will learn that force is a vector. A vector is a quantity that includes
information about the size, strength and direction.
What I Need to Know
At the end of this module, you should be able to:
1. Differentiate vector and scalar quantities (STEM_GP12V-Ia-8);
2. Perform addition of vectors (STEM_GP12V-Ia-9);
3. Rewrite a vector in component form (STEM_GP12V-Ia-10)
What’s In
There are physical quantities in physics which are represented by magnitudes and
units. When we say that the mass of an umbrella is 3 kilograms, such information gives the
magnitude and unit of mass. This quantity is called scalar. Mass therefore is an example of
scalar quantity.
Other quantities however cannot be completely specified by a magnitude and unit
alone. To describe the velocity of a car by saying that it is 50 kmh is incomplete. There is still
a need to describe the motion of direction of the car. The type of quantities which contains
direction is called the vector.
What’s New
Since vectors are quantities with specified magnitudes and direction, the most
appropriate representation of the vectors is the arrow (
). Why? This is because the
length of the arrow with respect to some chosen scale indicates the magnitude while the
arrowhead represents the direction of the vector. A vector is usually represented by a letter
with an arrow above it ( A ) .The same letter without an arrow indicates the magnitude.
What you will do?
ACTIVITY 1: Graphical Representation of Vectors
You will need
Ruler protractor pencil meterstick
Procedure
1. Walk around the four corners of your classroom. Record the direction of your
paths.
2. Measure the lengths and the widths of the room in meters.
3. Record the measured lengths and widths with the specified direction.
4. Represent the quantities graphically by using a convenient scale.
Questions:
1. What is the length? Width?
2. Give the direction of your path while walking along the lengths and widths of the
room.
3. Represent your results graphically using a specified scale
What Is It
A scalar is a quantity that can be completely described by a single value called
magnitude. Magnitude means the size or amount and always includes units of
measurement. Sometimes a single number does not include enough information to
describe a measurement. Giving complete directions would mean including instructions to
go to two kilometres to the north, turn right, then go to two kilometres east. The information
“Two kilometres to the north” is an example of a vector.(p110 Tom Hsu, Ph.D.)
The direction of some vectors is given in terms used by weather forecaster,
travellers, map readers and cartographers. The basic reference for angles is the following
N
W
N
S
E
W
E
North
South
East
West
NE
NW
SW
SE
Northeast
Northwest
Southwest
Southeast
S
For example: we draw a vector for a wind blowing at 30km/h in the northeast direction 450.
N
30km/h
45o
W
E
S
2
SCALAR ADDITION
Adding scalar quantities is similar to ordinary addition. We add together the
quantities express in the same units. Look at the example.
Example:
If the mass (m1)=25g and another mass (m2)=50g their sum is m=m1 + m2
Thus,
m= 25g + 50g
m= 75g
However, you must be careful about how the given magnitude are expressed. If there
are two different units, you need to convert the unit before adding.
Example:
If mass (m1)= 25g and another mass (m2)= 5kg. What is the total mass of an object?
Thus, convert g to kg so that,
25g
x 1kg/1000g =25/1000 or 0.025kg
Therefore, m = m1 + m2
=0.025 kg + 5kg
=5.025kg (total mass of an object)
VECTOR ADDITION
Now that we know how to represent vectors graphically, we are now ready to add two
or more vectors.
For example, we are asked to determine the vector sum of the resultant of the
following vectors:
1. A= 2 units, East
2. A= 2 units, East
B= 4 units, East
B= 4 units, West
Solution:
2 units
4 units
2 units
4 units
2 units West
6 units East
(R) Resultant
( R) Resultant
NOTE:
To determine the resultant of two or more vectors acting together in the same
direction, add the given vectors and take the common direction. On the other hand, to find
the resultant of two vectors with opposite direction, get the difference and take the direction
of the vector with the greater value.
What’s More
ACTIVITY 2: ( Graphing of vectors). Represent the following vectors graphically:
A: 40 units, 45o North of East
B: 50 units, 30o South of West
C: 40 units, 45o Counterclockwise from the (+) x-axis.
D: 50 units, 30o clockwise from the (-) y-axis
ACTIVITY 3: (Scalar addition) In separate paper, solve the following problem with
complete solution.
1. If area A1=20cm2 and area A2=36cm2, the total is A= A1 + A2
2. If the mass (m1) = 15kg and the second mass (m2) = 250g. Find the total mass
m= m1 + m2. Convert kg to grams.
ACTIVITY 4. (Vector Addition) In separate paper. Solve the given vectors and find the
resultant with exact direction.
1. A= 7 units, East
3. A= 2 units, East
B= 3 units, East
B= 8 units, West
2. A= 8 units, East
4. A= 3 units, East
B= 4 units, West
B= 9 units, West
Lesson
RESULTANT OF VECTORS
2
What’s In
When you draw a force vector on a graph, distance along the x or y- axes represents
the strength of the force in the x- and y- directions. A force at an angle has the same effect
as two smaller forces aligned with the x- and y- direction. To determine the resultant of two
or more vectors acting together in the same direction, add the given vectors and take the
common direction.
What I Need to Know
At the end of this lesson, you should be able to:
1. Determine the vector sum of two or more vectors;
2. Apply the Pythagorean theorem formula in getting the resultant vector;
3. Determine the vector resultant using the component method
4
What Is It
RESULTANT OF VECTORS
How do you determine the resultant of vectors acting at certain angles? The following
are the different methods which are used to determine the vector sum or resultant of two or
more vectors.
1. Graphical Method
a. Parallelogram
b. Polygon method
2. Analytical or Mathematical method
a. Trigonometrical method
b. Component method
In the graphical method, the vectors to be added are drawn according to a
convenient scale. They also have to be drawn in their specified direction. These directions
are usually indicated by an angle measured from a certain reference line. With the aid of
protractor, vectors can be drawn in their respective direction and the resultant is then drawn.
If the vectors are represented by rectangular coordinate system, the angle is measured with
respect to the x-axis or the y-axis.
In the analytical method, the vectors need not be drawn according to scale. A rough
sketch is simply made to show their magnitude and direction. The figure is analysed
mathematically with the application of mathematical procedures and formulas.
Graphical Method (Parallelogram)
The parallelogram method is a common way of adding two vectors. For example,
vector A and vector B are two vectors to be added, they are drawn from a common point.
The angle Φ (phi) is included angle between vector A and vector B. A parallelogram is then
formed using the vectors A and B as two sides.
Φ
A
R
ΦΦ
Φ
Φ
Φ
0
B
The resultant R is the diagonal line of the parallelogram from the common point 0.
The magnitude of the resultant is obtained by using the length and the established scale.
The angle Φ (theta) relative to B is simply measured by a protractor.
Sample Problem: (Used scientific calculator )
Carlito was observing an ant crawled along a tabletop. With a piece of chalk, he
followed its path. He determine the ants displacements by using a ruler and protractor. The
displacements were as followed; 4cm east and change its direction 7cm 450 north of east.
R=11cm
7cm NE
R= 11cm North of East
450
0
4cm
E
5
The ant went 4cm east and change direction 7cm, 450 north of east (NE). The
resultant displacement does not change. The order in which displacement vectors
are taken does not affect the resultant (11cm NE).
Analytical Method (Trigonometrical)
We can also get the resultant of two vectors by analytical method that is the
method of trigonometry. In the case of two perpendicular vectors, the included angle
is equal to 900. The value of the resultant is determined by applying the Pythagorean
theorem. Thus;
R2 = A2 + B2
Example: What is the vector sum of 8cm and 5cm acting on point O at an angle of
600?
Solution:
R2 = A2 + B2 - 2AB cos (120)
R2 = (8cm)2 + (5cm)2 - 2(8cm) (5cm) (-0.5)
R2= 64cm2 + 25cm2 - 2(40cm2)(-0.5)
R2= 64cm2 + 25cm2 – 2 (-20cm2)
R2=64cm2 + 25cm2 + 40cm2
R2=129cm2
B=5cm
R= 11.36cm
2
R = √129𝑐𝑚
600
23.180 1200
R = 11.36cm (The resultant vector)
0
A = 8cm
To find the angle θ (theta) may be determined from the sine law;
R
B
=
sin θ
Sin 1200
11cm
5cm
=
sin θ
Sin 1200
(11cm)(sin θ)
(5cm)(sin 1200)
=
11cm
11cm
Sin θ
=
5 ( 0.866)
11
= 0.3936
Sin θ
θ
= arc sin 0.3936
θ
= 23.180
Analytical (Component method)
The component of a given vector makeup the set of vectors whose vector sum is the
given vector.
The following procedure are applied in adding several vectors in terms of their
components:
1. Resolve the initial vectors into their components in the x and y directions
6
2. Add the component in the x direction to give Rx and add the components in the y
direction to give Ry. The following formulas help explain the second procedure:
Rx = x- component of R
= Ax + Bx + Cx + ……
= sum of the x-components
Ry = y- component of R
= Ay + By + Cy + ……
= sum of the y-components
3. Find the magnitude and direction of the resultant R from the components Rx and Ry.
The Pythagorean theorem is used , Thus;
R = √𝑎2 + 𝑏 2
The direction of R can be found the values of the component by trigonometry.
This is given by:
Tan θ = Ry
Rx
or
θ = arc tan
Ry
Rx
The angle θ may be situated in any of the four quadrants depending on the directions
of Rx and Ry. There are four possible cases and those are shown below;
Rx Ry
1
2
3
4
How θ is measured
Position of R
st
+ +
- +
- + -
1 Quadrant
2nd Quadrant
3rd Quadrant
4th Quadrant
Example:
Given:
Counterclockwice from (+) x-axis
Clockwise from (-) x-axis
Counterclockwise from (-) x-axis
Clockwise from (+) x-axis
A = 10 units, 300 counterclockwise from (+) x-axis
B = 25 units, 450 clockwise from (-) x- axis
y
Solution:
B
A
450
300
x
The figure shows A and B relatives to the rectangular coordinate system
Ax = A cos 300
= 10 units (0.866)
= 8.66 units
Ay = A sin 300
= 10 units (0.5)
= 5 units
The component of B are similarly obtained;
= B cos 450
= 25 units (0.707)
= -17.68 units
This is considered negative since Bx is to the left
By = B sin 450
= 25 units (0.707)
= +17.68 units
Since By is upward, we consider it to be positive.
Bx
To summarize the components of A and B, the following data are given
x-component
y-component
Ax = 8.66 units
Ay = 5 units
Bx = -17.68 units
By = 17.68 units
Therefore, to determine Rx and Ry, we have
Rx = Ax + Bx
= 8.66 units + (-17.68 units)
= - 9.02 units
The negative sign indicate that Rx is directed to the left
Ry = Ay + By
= 5 units + 17.68 units
= +22.68 units
The positive sign indicates that Ry is directed upward
The magnitude of the resultant is obtained by using the following formula
R =√(𝑅𝑥 )2 + (𝑅𝑦 )2
=√(−9.02 𝑢𝑛𝑖𝑡𝑠)2 + (22.68 𝑢𝑛𝑖𝑡𝑠)2
= √81.36 𝑢𝑛𝑖𝑡𝑠 + 514 𝑢𝑛𝑖𝑡𝑠
= √595.74 𝑢𝑛𝑖𝑡𝑠
R = 24.41 units
The direction of the resultant is evaluated using the following formula
Tan θ =
𝑅𝑥
𝑅𝑦
θ= arctan
Ry
Rx
R
Ry
θ= arctan
22.68 units
-9.02 units
= arctan 2.51
68.310
θ = 68.31 measured clockwise from the (-) x-axis
0
0
Rx
What’s More
ACTIVITY 5: Graphical Method (Parallelogram). illustrate the given magnitude in the
problem. Use separate paper and scientific calculator
.
Problem:
A group of soldiers walked 15 km north from their camp, then covered 10 km
due east. Note: Scale ( 1cm=1km).
a. What was the total distance walked by the soldier?
b. Determine the total displacement and exact direction of the travel.
c. Give the clear illustration of the path.
ACTIVITY 6: ( Analytical Method/ Trigonometrical). Calculate the vector sum and
illustrate. Use separate paper and scientific calculator.
Problem:
What is the vector sum of 15cm East and 9cm North of east acting on point O
at an angle of 800?
Find: a. R resultant force
b. direction of R
c. angle of θ (theta)
ACTIVITY 7: ( Component method) Determine the Resultant of two vectors using the
component method and illustrate. Use separate paper and scientific
calculator.
Given:
A = 15 units, 25o Counterclockwise from (+) x-axis
B = 30 units, 40o clockwise from (-) x- axis
9
What I Have Learned
A. Use the tail to tip method to add the given vectors
1. C= 6 cm, due north
D= 8 cm, due west
2. A = 7 km, due east
B = 3 km, due west
3. C = 4 cm, due south
D = 3cm, due west
4. E = 10 km, northwest
F = 20 km, northeast
5. G = 50 mm, northwest
H = 5 mm, northwest
Assessment: (Post-Test)
Multiple Choice. Select the letter of the best answer from among the given choices.
Direction: Multiple choices. Select the letter that correspond to the best answer.
1. Which of the following statement is true in determining the resultant of two or more
vectors acting together in the same direction?
a. Add the given vectors and take the common direction
b. Subtract the given vectors and take the common direction
c. Get the product of the given vectors and take the two direction
d. Add the given vectors and take the two direction
2. All statement are true about vectors, except;
a. Vectors are quantities with specified magnitude and direction
b. Vectors are quantities that have magnitude only
c. Vectors of two or more can be add by component method
d. Vectors can be add and take the common direction
3. A quantity that can be completely described by a single value called magnitude.
a. Vector
b. magnitude
c. scalar
d.velocity
4. The following statement is an example of scalar, except;
a. An ant crawl on top of the table 15 cm.
b. The troop of soldier walking 20km to northward direction
c. A bunch of flowers weighing 5 kilograms
d. A car runs fast 50km/h
5. A quantity that have magnitude and direction.
a. Vector
b. scalar
c. displacement
d. velocity
6. What is the vector some of 3 unit east and 5 unit east?
a. 5 units east b. 6 units east
c. 7 units east
d. 8 units east
7. To find the resultant of two vectors with opposite direction, you need to;
a. Get the difference and take the direction of the vector with the greater value
b. Get the sum and take the direction of the vector with the greater value.
c. Get the product and take the direction of the vector with the greater value
d. Get the product and take the direction of the vector with the smaller value
8. Methods which are used to determine the vector sum or resultant of two or more vectors,
except;
a. Parallelogram
b. polygon
c. component
d. symmetrical
9. Which of the following is true about graphical method in calculating resultant vector?
a. The vectors to be added are drawn according to a convenient scale
b. The vectors to be subtracted are drawn according to a convenient scale
c. The product of vectors are drawn according to a convenient scale
d. The vectors to be subtracted are directly drawn.
10. Since vectors are quantities with specified magnitudes and direction, the most appropriate
representation of the vector is;
a. Arrow
b. line
c. curve
d. angle theta
11. The formula used in obtaining the magnitude of the resultant in component method.
a. Pythagorean theorem b. Polygon method c. arc tangent d. Trigonometric
12. Which of the following is an example of vector quantity?
a. 25km/h
b. jumping 20 seconds
c. 50 grams
d. 5cm, east
13. What is the difference of the two vectors 8 units, east and 5 units, west?
a. 3 units, west b. 3 units, east
c. 4units west d. 4units, east
14. What is the exact sum of vectors A= 10cm, east and B= 5cm, east?
a. 5 cm, east
b. 5 cm, west
c. 15cm, east
d.15cm, west
15. The net displacement obtained from two or more vectors.
a. Scalar
b. resultant
c. sum
d. force
11
Answer key
Pretest
Activity 4
1. 7units
1 .A
2. B
3. C
4. B
5. A
6. D
7. A
8. D
9. A
10. A
11.A
12. D
13.B
14. C
15. B
ACTIVITY 2
3units
10 units east
2. 8units
4 Units
4nits east
3. 2units
8units
6units west
3units
A.
9units
40units NE
6units west
45
0
Posttest
B.
300
50 units SW
C.
y
40 units
X
D.
50 units SW
1 .A
2. B
3. C
4. B
5. A
6. D
7. A
8. D
9. A
10. A
11.A
12. D
13.B
14. C
15. B