What I Need To Remember
• Accuracy is defined as the closeness of a measured value to a true
or accepted value (Santos 2017).
• Precision is the amount of consistency of independent
measurements and the reliability or reproducibility of the
measurements (Santos 2017).
• A measurement x is reported as 𝑥 ± ∆𝑥 where x is the best
estimate and ∆𝑥 is the uncertainty.
• Estimating a measurement in a set of measurements using the
standard deviation can be reported as:
𝑥 = 𝑥̅ ± 𝜎
where x is the mean of the set of measurements and σ is the
standard deviation of the measurement.
Lesson
3
Vectors and Vector Addition
What I Need To Know
At the end of this lesson, you are expected to:
o differentiate vector and scalar quantities;
o perform addition of vectors; and
o rewrite a vector in component form.
What’s In
”
You made it! This is now the last lesson of Module 1.
Before we
will start with the lesson proper, there are terms that you need to
I will guide you until the end of the lesson. Let’s recall
know
and remember
firstand
so that
you will
understand
functions
the important
terms
concepts
you’ve
learned
in the
previous lesson as shown below.
The accuracy of experimental values refers to the closeness of these values
to the true or accepted value of a physical quantity while its precision refers
to its closeness to one another.
In physical experiments, it is important to have a measurement of
uncertainty. Standard deviation provides a way to check the results as
shown below:
𝑥 = 𝑥̅ ± 𝜎
where x is the mean of the set of measurements and σ is the standard
deviation of the measurement.
What’s New
An airplane travels North at 900kph and then travels east at
800kph to reach a destination. What would be its speed if it
would travel directly Northeast at 41.63°?
𝜃
Image 17
Guide Questions:
1. What is its total distance covered in the first route? in the new
route?
2. What is its displacement following the first route? the second route?
3. What is the difference between distance and displacement?
What is It
SCALAR AND VECTOR QUANTITIES
Scalar quantities are those that have magnitude (size) but no
direction (Homer and Bowen-Jones 2014).
Vector quantities are those that have magnitude (size) and
direction (Homer and Bowen-Jones 2014).
In expressing physical quantities, consider the example shown below:
SCALAR, distance
x = 10 m
magnitude
unit
VECTOR, displacement
x = 10 m, East
direction
magnitude
unit
Few examples of scalar and vector quantities are shown in the table below.
Scalar Quantities
Vector Quantities
⃗
Mass (m)
1kg
100 N, upward
Force (𝐹 )
997 kg/m³
2 km, Northeast
Density (𝜌)
Displacement (𝑑⃗)
Speed (s)
60 km/h
Velocity (𝑣⃗)
40 km/h, west
Distance (d)
120 m
Acceleration (𝑎⃗)
9.8 m/𝑠 2 , south
𝑚
Time (t)
55 s
Momentum (p)
20kg 𝑠 , westward
Representing a Vector Quantity
A vector quantity is represented by a line with an arrow (Homer and Bowen-Jones
2014).
Parts
Direction of the
arrowhead
Length of the line
Tail
Head
Meaning
Illustration
indicates the direction of
the vector
represents the magnitude
length of
of the vector
the line
tells the starting point of
the vector
tip of the vector, the
pointed end of the arrow
(Geometric & Algebraic Representations of Vectors, 2017)
Since vectors involves directions, a guide is also shown below.
This PhotoImage
by Unknown
17
Author is licensed under CC
Image 18
BY-SA
where,
N – North
NE – East of North
E – East
SE – East of South
S – South
SW – West of South
W – West
NW – West of North
Adding Vector Quantities
In writing vectors, we use a single letter with an arrow above it. For example, vector
A is represented as 𝐴⃗.
Adding vectors would lead to another vector which is called the resultant
⃗⃗⃗⃗ is simply denoted 𝑎𝑠 𝐴⃗ + 𝐵
⃗⃗⃗⃗ considering the
vector. Vector addition of 𝐴⃗ and 𝐵
directions. The directions with north and east will have positive sign while south and
west will have negative.
⃗⃗⃗⃗. There are different
To get the resultant vector, we use the equation 𝑅⃗⃗ = 𝐴⃗ + 𝐵
methods in adding vectors. In this module, we will use two methods, (i) Graphical
Method/ Scale Drawing Approach and (ii) Analytical Method using the Component
Method.
Graphical Method/ Scale Drawing Approach
⃗⃗ which are not in the same direction
Adding two vectors 𝐴⃗ and 𝐵
can be done by forming a parallelogram to scale.
⃗⃗ below,
Look at vectors 𝐴⃗ and 𝐵
⃗⃗ using the graphical method or the scale drawing
To add vectors 𝐴⃗ and 𝐵
approach, refer to the steps below.
STEPS
ILLUSTRATION
Step 1. Make a rough sketch of how the vectors Given:
are going to add together to give you an idea of
how large your scale needs to be in order to fill
the space available to you.
Note:
- The scaling ratio depends on the given
measurements of the vectors. For example, 𝐴⃗ has the
⃗⃗ has the magnitude of 4
magnitude of 3 m and 𝐵
m. Then the scale ratio of 1 cm = 1 m can be used.
- For the angle measurement, a protractor must be
used.
Image 19
Step2: Having chosen a suitable case, draw the
⃗⃗ so that
scaled lines in the direction of 𝐴⃗ and 𝐵
they form two adjacent sides of the parallelogram.
Tip: Use the “head-to-tail” approach. Draw the first
vector and connect its head to the tail of the second
vector.
Step 3. Draw the remaining two sides to complete
the parallelogram.
* The broken lines represent the remaining two
sides of the parallelogram.
Step 4. The diagonal line, 𝑅⃗⃗ represents the
resultant vector in both magnitude and direction.
* The resultant arrow should start from the tail of
⃗⃗.
vector 𝐴⃗ and end at the head of the vector 𝐵
(Homer and Bowen-Jones 2014)
Example 1. Two forces of magnitude 4.0 N and 6.0 N act on a single point.
The forces make an angle of 60° with each other. Using a scale diagram,
determine the resultant force (Homer and Bowen-Jones 2014).
Solution:
A vector must have a magnitude and a direction which
means the angle is as important as the size of the force.
Scale: 10 mm = 1.0 N
6N
⃗⃗⃗ = 8.7 N
𝑹
𝜃 = 36°
4N
Image 20
The length of the resultant is 87 cm so the force is 8.7 N. The angle the resultant
makes with 4N force is 36° (Homer and Bowen-Jones 2014).
Analytical Method using Component Method
The word “component” means part. Hence, the components of a vector
mean the parts of a vector. A vector has an x-component and a ycomponent (Santos 2017).
Suppose a vector ⃗𝑨⃗ is on a Cartesian Coordinate system with its tail at the origin
and makes an arbitrary angle θ with the positive x axis as shown below.
(a)
(b)
⃗⃗ can be represented as a vector sum of vectors ⃗⃗⃗⃗⃗
In figure (a), vector 𝐀
𝐀 𝐱 and
⃗⃗⃗⃗⃗⃗
⃗
⃗
𝐀 𝐲 , known as component vectors of 𝐀. The subscripts x and y describe where
the vectors are lying in the plane. The lines parallel to the components are
equal in magnitude to the components as shown in figure (b) but the angle
is with respect to the y-axis.
To solve for the x and y components of a vector, we can apply the trigonometric
functions sine, cosine and tangent. Recall that a right triangle has sides A and B and
the hypotenuse C. With reference to an angle θ, side A is the adjacent side and side
B is the opposite side as shown below.
Recall SOH-CAH-TOA:
⃗⃗⃗⃗⃗𝒙 is side A, vector ⃗⃗⃗⃗⃗
⃗⃗ is our
Using the figure (a) above, vector 𝑨
𝑨𝒚 is side B and vector 𝑨
hypotenuse. Therefore, we can express the components of vector ⃗𝑨⃗ as:
Note:
Equations 1 and 2 are only true when the angle
θ is with respect to the x-axis. Identify correctly
and carefully the opposite and adjacent sides
based on the location of the reference angle.
Use the trigonometric functions presented
above to find the components of a given vector.
Resultant Vector
To compute for the result vector given the components, use Pythagorean Theorem.
𝟐
⃗𝑨⃗ = √⃗⃗⃗⃗⃗
𝑨𝒙 + ⃗⃗⃗⃗⃗
𝑨𝒚
𝟐
To calculate for the angle, the trigonometric function tangent is used.
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
tan 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
For figure (a), the angle can be calculated using the equation below.
𝜃 = 𝑡𝑎𝑛−1 (
⃗⃗⃗⃗⃗
𝑨𝒚
)
⃗⃗⃗⃗⃗
𝑨𝒙
Example 2. Ayah walks every day in going to school. From her house, she
walks 120 m, east and continues to walk another 100 m in the direction of
30° south of east to reach her destination. Determine the magnitude and
direction of Ayah’s resultant displacement.
Solution:
Step 1: Assign
vectors for each
displacement.
Vector1 is 120 m, east
Vector2 is 100 m, 30° south of east
Step 2: Draw the
vectors
Note:Every vector
will be placed in a
Cartesian
Coordinate plane
with its tail at the
origin.
Step 3: Draw the x
and y-components
of each vector.
Note: Since V1 =
120 m, East, there
is no y-component.
Step 4: Solve for the
x and ycomponents of each
vector.
⃗⃗⃗⃗𝑥
𝑉
⃗⃗⃗⃗
𝑉𝑦
⃗⃗⃗⃗
𝑉1
⃗⃗⃗⃗⃗⃗
𝑉
1𝑥 =120 m
(+, East)
No ycomponent
⃗⃗⃗⃗
𝑉2
CAH
cos 𝜃 =
⃗⃗⃗⃗⃗⃗⃗⃗
𝑉2𝑥
⃗⃗⃗⃗⃗
𝑉2
⃗⃗⃗⃗⃗⃗⃗⃗
𝑉2𝑥
cos 30 ° =
100
⃗⃗⃗⃗⃗⃗⃗
𝑉
2𝑥 = 100cos30°
⃗⃗⃗⃗⃗⃗⃗
𝑉
2𝑥 =86.60 m
(+, East)
SOH
sin 𝜃 =
⃗⃗⃗⃗⃗⃗⃗⃗
𝑉2𝑌
𝑅⃗⃗
⃗⃗⃗⃗⃗
𝑅𝑥 = 120 m +
86.60 m
⃗⃗⃗⃗⃗
𝑅𝑥 = 206.60 m
⃗⃗⃗⃗⃗
𝑅𝑦 = -50 m
⃗⃗⃗⃗⃗
𝑉2
⃗⃗⃗⃗⃗⃗⃗⃗
𝑉2𝑦
sin 30 ° =
100
⃗⃗⃗⃗⃗⃗⃗
𝑉
=
100sin30°
2𝑥
⃗⃗⃗⃗⃗⃗⃗
𝑉
2𝑥 = −50 m
(-, South)
Step 5: Solve for the Use Pythagorean Theorem,
magnitude of the
𝑅⃗⃗ = √(𝑅⃗⃗𝑥 )2 + (𝑅⃗⃗𝑦 )2
resultant vector, 𝑅⃗⃗
𝑅⃗⃗ = √(206.60 𝑚)2 + (−50 𝑚)2
𝑅⃗⃗ = √42,683.56 𝑚2 + 2,500 𝑚2
𝑅⃗⃗ = 212.56 𝑚
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
Step 5: Solve for the
tan 𝜃 =
angle 𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
⃗⃗⃗⃗⃗⃗
𝑅
𝑦
tan 𝜃 = ⃗⃗⃗⃗⃗⃗
𝑅
𝑥
−50 𝑚
Final Answer:
tan 𝜃 = 206.60 m
𝜃 = 13.60° south of east
⃗⃗
𝑅 = 212.56 𝑚, 13.60° south of east