1. Methods of Finding Vector Sum
Phys 13 General Physics 1
Methods of Finding Vector Sum
MARLON FLORES SACEDON

2. Vector analysis Graphical Method Analytical Method
Methods of finding Vector sum or resultant of forces
Parallelogram Method
Polygon Method
Graphical Method
Cosine law Method
Component Method
Analytical Method

3. Vector analysis Two forces Graphical Method Parallelogram Method 𝑅𝜃 𝛽 𝛽
𝐹 2
𝐹′ 2 𝜃
𝐹 1
𝐹′ 1 𝜃 𝛽
Two forces
(Drawn to scale)

4. Vector analysis Two forces Graphical Method Polygon Method 𝐹 2 𝐹 1𝜃 𝛽
Two forces
(Drawn to scale)

5. Vector analysis Two forces Graphical Method Polygon Method 𝑅 𝐹 2 𝐹 2𝛽 𝜃
𝐹 1
𝐹 1 𝜃 𝛽 𝜃
Two forces
(Drawn to scale)
This is Polygon method

6. Vector analysis Two forces Analytical Method Cosine law Method 𝑅 𝐹 2𝜃 𝛽
𝐹 2
𝐹 1
Equivalent polygon of vectors
(Drawn not to scale))
𝐹 1 𝛽 𝜃
𝐹 2
Two forces
Drawn not to scale
Apply law sines to solve for 𝛼:
𝑠𝑖𝑛𝛼 𝐹 2 = 𝑠𝑖𝑛 𝜃+𝛽 𝑅 𝛼 𝛾
Magnitude of the Resultant R (apply law of cosines)
𝑅 or 𝑅= 𝐹 𝐹 2 2 −2 𝐹 1 𝐹 2 cos⁡(𝜃+𝛽)
Direction of the Resultant R
𝛾= 180 𝑜 −𝜃−𝛼

7. Vector analysis Analytical Method Component Method
Vector can be resolve in
x & y components
+ 𝐴 𝑦 𝐴 𝜃
+ 𝐴 𝑥

8. Vector analysis Analytical Method Component Method
Vector can be resolve in
x & y components
Components and its vector can be formed into a right triangle.
+ 𝐴 𝑦 𝜃 𝐴
+ 𝐴 𝑦 𝐴
𝑨 = 𝑨 𝒙 + 𝑨 𝒚
Resolution
of vectors
Composition
of vectors 𝜃
+ 𝐴 𝑥
+ 𝐴 𝑥
From right triangle (left side figure)
𝐴= 𝐴 𝑥 𝐴 𝑦 2
𝑐𝑜𝑠𝜃= 𝐴 𝑥 𝐴
𝐴 𝑥 =𝐴𝑐𝑜𝑠𝜃
𝜃= 𝑇𝑎𝑛 −1 𝐴 𝑦 𝐴 𝑥
𝑠𝑖𝑛𝜃= 𝐴 𝑦 𝐴
𝐴 𝑦 =𝐴𝑠𝑖𝑛𝜃

9. Vector analysis Two forces or more Analytical Method Component Method
Steps in Component Method
Step 2: Sum up all components along x-axis and all components along y-axis algebraically.
Step 1: Resolve the vectors into each components and solve their values using 𝐴 𝑥 =𝐴𝑐𝑜𝑠𝜃 &
𝐴 𝑦 =𝐴𝑠𝑖𝑛𝜃.
𝐹 𝑖𝑥 = 𝐹 𝑖 𝑐𝑜𝑠𝜃
From the Figure (left) we have, 𝛽
𝐹 2
+ 𝐹 2𝑦
+ 𝐹 1𝑦
𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +…
𝐹 1 𝜃
𝐹 𝑖𝑦 = 𝐹 𝑖 𝑠𝑖𝑛𝜃
Components of Resultant 𝑅
along x & y axis
𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +…
Step 3: Calculate the magnitude and direction of the resultant using 𝑅 𝑥 and 𝑅 𝑦 .
− 𝐹 1𝑥
𝐹 2𝑥
𝑅= Σ𝑅 𝑥 Σ𝑅 𝑦 2
Magnitude of resultant
Two forces or more
Drawn not to scale
𝛾= 𝑇𝑎𝑛 −1 𝑅 𝑦 𝑅 𝑥
direction of resultant

10. Vector analysis Steps in Component Method + 𝐹 1𝑦 𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +…
Step 1: Resolve the vectors into each components and solve their values using 𝐴 𝑥 =𝐴𝑐𝑜𝑠𝜃 & 𝐴 𝑦 =𝐴𝑠𝑖𝑛𝜃.
Step 2: Sum up all components along x-axis and all components along y-axis algebraically.
From the Figure (left) we have, 𝛽
𝐹 2
+ 𝐹 1𝑦 𝑹
𝑅 𝑥 =− 𝐹 1𝑥 + 𝐹 2𝑥 +…
𝐹 1 𝜃
𝑅 𝑦 = 𝐹 1𝑦 + 𝐹 2𝑦 +…
Step 3: Calculate the magnitude and direction of the resultant using 𝑅 𝑥 and 𝑅 𝑦 .
− 𝐹 1𝑥
+ 𝐹 2𝑦
𝑅= Σ𝑅 𝑥 Σ𝑅 𝑥 2
Magnitude of resultant
𝛾= 𝑇𝑎𝑛 −1 𝑅 𝑦 𝑅 𝑥
direction of resultant 𝛾
𝐹 2𝑥

11. Vector analysis Problem Find the magnitude and direction of resultant
using polygon method, cosine law and component method.
200𝑔𝑓
30 𝑜
65 𝑜
500𝑔𝑓
35 𝑜
400𝑔𝑓

12. Vector analysis 𝜸 Application Problem
end
Displacement (D)
𝐷 𝑦
8 km
Application Problem
A cross-country skier skis 5.00 km in the direction 500 east of south, then 3.00 km in the direction N 600 E, and finally 8.00 km with bearing angle of Find the displacement of the skier. N E S W 𝜸
Solving for the x component of displacement
=2.82 𝑘𝑚
𝐷 𝑥
start
𝐷 𝑥 =
+5 cos 50 𝑜
50o
+ 3 cos 30 𝑜
−8 cos 68 𝑜
5 km
Solving for the y component of displacement N E S W
𝐷 𝑦 =
− 5 sin 50 𝑜
+ 3 sin 30 𝑜
+ 8 sin 68 𝑜
=5.09 𝑘𝑚
68o
338o
Solving for the magnitude and direction of displacement
3 km N E S W
60o
𝐷= =5.82 𝑘𝑚
30o
𝛾= 𝑇𝑎𝑛 − = 61 𝑜
East of North

13. eNd