scalar and vector

1. Scalar and Vector
Week 2 Lesson 1
12th grade

2. Table of contents
01.
02. 04.
03.
Scalar and Vector
Vector Resolution
using Component
Method
Vector Resolution
using
Pythagorean
Theorem and
Tangent Function
Vector
Representation
and Addition

3. Quote for Today!

4. Scalar and Vector
01.

7. Vector
Representation
02.

8. How to read and
draw Vectors?
● Vectors are illustrated using a straight
arrow. The tail is the origin while the head is
the terminal.
● Vector arrows should be placed on the
cartesian plane to illustrate the direction
which can be in cardinal or polar form.
● Cardinal Directions are North, West, South
and East.
● Polar Directions are in terms of angles.

9. Let’s say that each square is equal to 1
units, How can we read Vector 𝐴?
𝐴 = 20 units, 30o North of east
𝐴 = 20 units, 60o East of North (NE)
𝐴 = 20 units, -3300
30o
30o 𝐵 = 10 units, 60o North of east
𝐵 = 10 units, 30o East of North (NE)
𝐵 = 10 units, -3000

10. Adding vectors can be resolved geometrically
because of their directions. The sum of the
vectors is commonly known as Resultant Vector.
It can be expressed as:
Vector Addition
𝑅 = 𝐴 + 𝐵 + …. +
𝑛
The Resultant vector can be determined by
Graphical Solution and Analytical Solution

11. Graphical Solution
Known as Tail-Tail Method.
Both tails of the two vectors
are connected at the origin of
the cartesian plane. At each
head of the vectors, draw line
parallel to the other
Known as Tail-Head Method.
Connect the tail of the first
vector at the origin of the
cartesian plane and the
succeding tails of vectors are
connected at the head of the
last vector drawn.
Polygon Method
Parallelogram
Method

12. Polygon Method
Sample Problem: Find the Resultant of the two forces acting on an object by
using polygon method: 𝐴 = 10.0 N, 20o west of south and 𝐵 = 16.0 N, east.
(1 cm = 2N)
Given: 𝐴 = 10.0 N, 20o west of south
𝐵 = 16.0 N, east.
Find: Resultant vector using polygon method

13. Polygon Method
Solution: Using a scale 2 N = 1 cm, scaled magnitude 𝐴 and 𝐵 should be
drawn on cartesian plane
1. Prepare graphing paper, marker or pencil and protractor. Draw cartesian plain.
2. Draw the first vector. Using a protractor, locate the 20o west of south. Put a
mark for the arrow representation of vector 𝐴.
3. Draw the 5 cm length as the scaled magnitude of vector 𝐴 from the origin of
the cartesian plane.
4. Label as vector 𝐴.
5. Connect vector 𝐵 from the head of the vector following the same procedures
of the previous vector.

14. Polygon Method
Solution: Using a scale 1 N = 1 cm, scaled magnitude 𝐴 and 𝐵 should be
drawn on cartesian plane
6. Draw the Resultant vector from the origin of the first vector to the head of the last
vector.
7. The Resultant vector if in the fourth quadrant. Measure the angle within the
quadrant.
8. Measure the length of the Resultant vector.
9. Write the Resultant vector.

15. 5 cm
𝐴
𝐵
200
8 cm
𝑅
𝑅 = 7.8 cm = 15.6 N
𝑅 = 15.6 N, 530 east of south
𝑅 = 15.6 N, 370 south of east
𝑅 = 15.6 N, 3230
𝑅 = 15.6 N, -370
7.8 cm

16. Parallelogram Method
Sample Problem: Find the Resultant of the two forces acting on an object by
using polygon method: 𝐴 = 10.0 N, 20o west of south and 𝐵 = 16.0 N, east.
(1 cm = 2N)
Given: 𝐴 = 10.0 N, 20o west of south
𝐵 = 16.0 N, east.
Find: Resultant vector using parallelogram method

17. Parallelogram Method
1. Construct the two vectors in the same origin of a cartesian plane using protractor
and ruler
2. At the head of each vector, draw a line parallel to each other.
3. Draw the resultant vector from the origin to the point where parallel lines
intersect and measure the length and the angle.
4. Write the Resultant vector.

18. 𝐴
𝐵
200
𝑅
𝑅 = 7.8 cm = 15.6 N
𝑅 = 15.6 N, 530 east of south
𝑅 = 15.6 N, 370 south of east
𝑅 = 15.6 N, 3230
𝑅 = 15.6 N, -370
7.8 cm

19. Clarissa walks 9.0 m to the east and then runs 12 m in
a direction of 750 south of east. Find the resultant
vector using polygon and parallelogram method.
3m = 1 cm

20. Analytical Solution
Is done by taking each part of
a vector along the axes of a
cartesian plane.
Solving for components is
done by using sine and cosine
function;
Finding the magnitude and the
angle of the resultant vector is
done using the Pythagorean
theorem and the Tangent
Function, respectively
Is used when the vectors are
connected tail to head, and
with the resultant vector will
form triangle.
If the triangle formed is a right
triangle, use Pythagorean
Theorem and the Tangent
function;
If the traingle formed is an
acute or an obtuse triangle,
use law of sine and cosine.
Triangle Method Component Method

21. Pythagorean Theorem
Tangent Functions
To find
Magnitude
To determine
Direction

22. Sample Problem: Find the Resultant of the two forces acting on an object by
using Triangle method: 𝐴 = 10.0 N, 20o west of south and 𝐵 = 16.0 N, east.
(1 cm = 2N)
Given: 𝐴 = 10.0 N, 20o west of south
𝐵 = 16.0 N, east.
Find: Resultant vector using triangle method
Triangle Method
𝐴
𝐵
200
𝑅

23. The angle between 𝐴 and 𝐵 is 700. (Acute
angle) Use the cosine law to determine
magnitude of 𝑹.
𝑅 = 𝐴2 + 𝐵2 − 2𝐴𝐵𝑐𝑜𝑠700
𝑅 = (10.0𝑁)2+(16.0𝑁)2 − 2(10.0𝑁)(16.0𝑁)𝑐𝑜𝑠700
𝑅 = 100 + 256 − 320 (𝑐𝑜𝑠700)= 246.55𝑁
𝑅 = 15.7 N
To locate the direction, compute first the angle between 𝑨
and 𝑹 using the sine law.
Triangle Method
𝐴
𝐵
𝑅
𝑅
sin 700 =
𝐵
sin 𝜃
𝜃 = 𝑠𝑖𝑛−1
𝐵𝑠𝑖𝑛700
𝑅
𝜃 = 𝑠𝑖𝑛−1
(16.0𝑁)𝑠𝑖𝑛700
15.7𝑁
𝜃 = 73.30
− 200
𝑅 = 15.7 N, 53.30 east of south

24. Sample Problem: Find the Resultant of the two forces acting on an object by
using Component method: 𝐴 = 10.0 N, 20o west of south and 𝐵 = 16.0 N, east.
(1 cm = 2N)
Given: 𝐴 = 10.0 N, 20o west of south
𝐵 = 16.0 N, east.
Find: Resultant vector using component method
Component Method
𝐴
𝐵
200
𝑅

25. Compute the components of each vector.
𝐴 Component
𝐴𝑥 = A sin θ
𝐴𝑥 = (10.0 N) sin 200
𝐴𝑥 = 3.42 N (since direction is west of south) = -3.42 N
𝐴𝑦 = A cos θ
𝐴𝑦 = (10.0 N) cos 200
𝐴𝑦 = 9.40 N (since direction is west of south) = -9.40 N
Component Method
𝐴
𝐵
𝑅
𝐵 Component
𝐵𝑥 = 16.0 N
𝐵𝑦 = 0

26. Solve for the Resultant component.
𝑅 Component
𝑅𝑥 = Ax + Bx
𝑅𝑥 = (-3.42 N) + (16.0 N)
𝑅𝑥 = 12.58 N
𝑅𝑦 = Ay + By
𝑅𝑦 = (-9.40 N) + (0)
𝑅𝑦 = -9.40 N
Component Method
𝐴
𝐵
𝑅
Solve for the Resultant magnitude angle θ.
𝑅 = 𝑅𝑥2 + 𝑅𝑦2
𝑅 = (12.58 𝑁)2+(−9.40 𝑁)2
𝑅 = 158.26 𝑁 + 88.36 𝑁
𝑅 = 246.62
𝑅 = 15.7 N
tan θ =
𝑅𝑦
𝑅𝑥
tan θ =
−9.40 𝑁
12.58 𝑁
Θ = tan-1 -0.747 N
Θ = -36.750 or 36.80 south of east
𝑅 = 15.7 N, 36.8 south of east
𝑅 = 15.7 N, 53.2 east of south
𝑅 = 15.7 N, -36.80
𝑅 = 15.7 N, 323.20

27. Clarissa walks 9.0 m to the east and then runs 12 m in
a direction of 750 south of east. Find the resultant
vector using triangle and component method.

28. Asynchronous Task
Answer the following in your notebook.
1. A dog, in following a lizard, goes 5 m east, turns 4 m north, and then goes 10 m
west. Find the magnitude and the direction of the dog’s resultant displacement
using the graphical method.
2. Using a convenient scale, find the resultant of these vectors graphically: 16 m at
1400 with the positive x-axis and 10 m at 300 with the positive x-axis.
3. Using the component method, find the resultant of these vectors: 8 N along the
positive x-axis and 6 N making an angle 450 with the x-axis.
4. An object is moved 8.0 cm towards the positive x-axis and then 6.0 cm at the
angle of 60 x-axis. Find the resultant displacement.
5. A 100-N force is making an angle of 600 with the horizontal. Find the vertical
and horizontal components of the force.

29. Performance Task

30. Resultant-Equilibrant
Objective:
Determine the resultant of concurrent vector forces
Materials:
2 spring balances
1 100-g mass
2 iron stand
2 strong string
Protractor
Ruler
Pencil and pen
Graphing paper
Lab Activity Sheet
3-5 pcs Short Bond Paper