The mathematics of vectors is quite different from the mathematics of scalar quantities. For Example, in the multiplication of scalar quantities we use the “dot product”, whereas in the multiplication of vector quantities we use the “cross product” or the vector product method to account for direction. Copy the link given below and paste it in new browser window to get more information on Parallelogram Law Force:- http://www.transtutors.com/homework-help/civil-engineering/fundamental-concepts/parallelogram-law-force.aspx

1. Parallelogram Law of
Forces
An introduction

2. What is Parallelogram Law?
An identity stating relation between sides and diagonals of a
parallelogram.
(𝑨𝑩) 𝟐+(𝑩𝑪) 𝟐+(𝑪𝑫) 𝟐+(𝑨𝑫) 𝟐= (𝑨𝑪) 𝟐+(𝑩𝑫) 𝟐
Sum of the squares of sides of
parallelogram is equal to the sum of
squares of diagonals of parallelogram.

3. Parallelogram Law of Vector
Addition
Modification of Parallelogram Law to facilitate addition of
vector quantities.
𝑨 2
+ 𝑩 2
+ 2 × 𝑨 × 𝑩 cos 𝜃 = 𝑸𝑺 2
𝑨 + 𝑩 = 𝑸𝑺
𝜽

4. Parallelogram Law of Forces
Application of Parallelogram Law of Vector Addition.
𝐹1
2
+ 𝐹2
2
+ 2 × 𝐹1 × 𝐹2 cos 𝜃 = 𝑸𝑺 2
𝑭 𝟏 + 𝑭 𝟐 = 𝑭

5. Special Cases
Case 1
• 𝜃 = 0°
• cos 𝜃 = 1
• sin 𝜃 = 0
• 𝑭 = 𝑭 𝟏 + 𝑭 𝟐
• 𝜶 = 𝟎°
Case 2
• 𝜃 = 180°
• cos 𝜃 = −1
• sin 𝜃 = 0
• 𝑭 = (𝑭 𝟏 ± 𝑭 𝟐) 𝟐
• 𝜶 = 𝟎° 𝒐𝒓 𝟏𝟖𝟎°
Case 3
• 𝜃 = 90°
• cos 𝜃 = 0
• sin 𝜃 = 1
• 𝑭 = 𝑭 𝟏
𝟐
+ 𝑭 𝟐
𝟐
• 𝜶 = 𝐭𝐚𝐧−𝟏(
𝑭 𝟐
𝑭 𝟏
)

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