1. Department of Mathematics & Physics
General Physics 1
Vector Analysis
Prof. MARLON FLORES SACEDON
Department of Mathematics & Physics

2. Vector analysis Scalar Vector Physical Quantity – e.g. Time (t)
, temperature (T)
, Mass (m)
, distance (s)
, density (ρ)
– specified by their magnitude only
Vector
– e.g. Force ( 𝐹 )
, Electric field ( 𝐸 )
, weight ( 𝑤 )
, displacement ( 𝑠 )
– specified by their magnitude and direction

3. Vector analysis Scalar Vector Physical Quantity
are added by ordinary algebraic method
– specified by their magnitude only
Vector
are added by geometric method
– specified by their magnitude and direction

4. Vector analysis Vector addition or vector sum 𝑃 2 𝑃 1
𝑃 3
Direction
𝑃 4
Total distance
𝑃 2
magnitude
𝑃 2
𝑃 5
Total Displacement
𝑃 1
𝑃 1
What is the total distance and
total displacement from 𝑃 1 to 𝑃 5 ?

5. Vector analysis = V 𝑉 𝐴 = 𝑉 𝐵 =− 𝐴 Vector addition or vector sum
Note: The magnitude of vector 𝑉 is 𝑉 𝑜𝑟 𝑉
𝑃 2
𝑃 4
𝑃 5
Vector 𝑉
= V 𝐴
= 𝑉 𝐵
=− 𝐴
𝑃 1
𝑃 3
𝑃 6
If vector V and vector A have the same magnitude, then the two vectors are equal.
If vector B and vector A have the same magnitude but in opposite direction, then the two vectors are not equal.

6. Vector analysis + 𝐵 + 𝐴 − 𝐶 =0 𝐴 𝐵 𝐶 𝐴 + 𝐵 = 𝐶 𝐶 = 𝐴 + 𝐵
Vector addition or vector sum
Parallel
vectors
Anti-parallel
vectors
+ 𝐵
+ 𝐴
− 𝐶 =0 𝐴 𝐵 𝐶
𝐴 + 𝐵 = 𝐶
This process is called Vector Algebra
𝐶 = 𝐴 + 𝐵
Vector Sum or Resultant of vectors A and B
Expressing vector 𝐶 in terms of 𝐴 and 𝐵
NOTE: “In a close polygon the vector sum is zero.”

7. Vector analysis In Figure (above), expressed the following vectors.𝑭 𝑫 𝑪 𝑨 𝑮 𝑬 𝑩 𝑲 𝑯
Figure
Example
In Figure (above), expressed the following vectors.
a) vector 𝑪 in terms of vectors 𝑬 , 𝑫 , and 𝑭 .
+ 𝑪
+ 𝑫
− 𝑬
+ 𝑭
+ 𝑬
− 𝑫
− 𝑭 =𝟎

8. Vector analysis In Figure (above), expressed the following vectors.𝑭 𝑫 𝑪 𝑨 𝑮 𝑬 𝑩 𝑲 𝑯
Figure
Example
In Figure (above), expressed the following vectors.
a) vector 𝑪 in terms of vectors 𝑬 , 𝑫 , and 𝑭 .
𝑪 = 𝑬 − 𝑫 − 𝑭 ANSWER

9. Vector analysis In Figure (above), expressed the following vectors.𝑭 𝑫 𝑪 𝑨 𝑮 𝑬 𝑩 𝑲 𝑯
Figure
Example
In Figure (above), expressed the following vectors.
a) vector 𝑪 in terms of vectors 𝑬 , 𝑫 , and 𝑭 .
+ 𝑪
+ 𝑫
− 𝑬
+ 𝑭 =𝟎
𝑪 = 𝑬 − 𝑫 − 𝑭 ANSWER
b) vector 𝑮 in terms of 𝑪 , 𝑫 , 𝑬 , and 𝑲 .
+ 𝑮
+ 𝑬
− 𝑫
− 𝑪
+ 𝑲 =𝟎
𝑮 = 𝑪 + 𝑫 – 𝑬 – 𝑲 ANSWER

10. Vector analysis Graphical Method Analytical Method
Methods of finding Vector sum or resultant of forces
Parallelogram Method
Polygon Method
Graphical Method
Material needed:
Triangular scale or ruler
Protractor
Writing paper and pencil
Cosine law Method
Component Method
Analytical Method
Material needed:
calculator
Writing paper and pen

11. 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)

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

13. 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
(Drawn to scale)

14. 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

15. to be continued…