Laws of Addition/Subtraction of Real Vectors

1. Addition of 2 Vectors or Subtraction of 1 Vector from Other have Special Geometric Interpretaion if the 2 Vectors either Have or are Brought to Having the following configuration
   1. The 2 Vectors are in Continuation with each other i.e. Head Point of one Vector is the Tail Point of the Other Vector. The Vector containing the Common Point as Tail is called the Leading Vector. The Vector containing the Common Point as Head is called the Trailing Vector.
   2. The 2 Vectors Share a Common Tail Point.
   3. The 2 Vectors Share a Common Head Point.
   If any 2 Vectors are brought under one of the above 3 configurations, the resultant vector of their Addition (or Subtraction) can be can be calculated using either the Triangular Law of Vector Addition OR the Parallelogram law of Vector Addition.
2. Triangular Law of Vector Addition states that If 2 Vectors that are in Continuation ( i.e. Head Point of one Vector is the Tail Point of the Other Vector) are added then the Resultant Vector has the Head Point of the Lead Vector and Tail Point of the Trailing Vector. Since the 2 Vectors and the Resultant Vector form a Triangle, it is know as the Triangular Law of Vector Addition.
   
   From Fig.1 as per Triangle Law of Vectors
   
   \(\vec{AC}=\vec{AB} + \vec{BC} = \vec{BC} + \vec{AB}\)
   
   And as per Cosine Law of Triangles
   
   \(|\vec{AC}|=|\vec{CA}|=\sqrt{|\vec{AB}|^2 + |\vec{BC}|^2 - 2|\vec{AB}||\vec{BC}|\cos \theta}\)
   
   Triangle Law can be used to Find the Resultant Vector under following conditions
   1. Addition of 2 Vectors that are in Continuation with each other: The resultant Vector of addition of 2 Given Vectors \(\vec{AB}\) and \(\vec{BC}\) is given as
      
      \(\vec{AB} + \vec{BC} = \vec{BC} + \vec{AB}= \vec{AC}\)
   2. Subtraction of 2 Vectors that Have a Common Tail Point: The resultant Vector of Subtration of 2 Vectors \(\vec{AB}\) and \(\vec{CB}\) is given as
      
      \(\vec{AB} - \vec{CB} = \vec{AB} + \vec{BC} = \vec{AC}\)
      
      \(\vec{CB} - \vec{AB} = \vec{CB} + \vec{BA} = \vec{CA}\)
   3. Subtraction of 2 Vectors that Have a Common Head Point: The resultant Vector of Subtration of 2 Vectors \(\vec{AB}\) and \(\vec{AC}\) is given as
      
      \(\vec{AB} - \vec{AC} = \vec{AB} + \vec{CA} = \vec{CA} + \vec{AB} = \vec{CB}\)
      
      \(\vec{AC} - \vec{AB} = \vec{AC} + \vec{BA} = \vec{BA} + \vec{AC} = \vec{BC}\)
   Polygonal Law of Vector Addition (which is an extension of the Triangle Law) states that If \(N\) Vectors that are in Continuation (such that Head Point of Any Vector is the Tail Point of the Next Vector) are added then the Resultant Vector has the Head Point of the Last Vector and Tail Point of the First Vector. Since the Vectors that are added and the Resultant Vector form a Polygon, it is know as the Polygonal Law of Vector Addition.
   
   From Fig.2 as per Polygon Law of Vectors
   
   \(\vec{AG}=\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} + \vec{EF} + \vec{FG}\)
   
   
3. Parallelogram Law of Vector Addition states that If 2 Vectors that Share a Common Tail Point are added then the Resultant Vector is given by the Diagonal of the Parallelogram whose 2 Adjacent Sides are given by the 2 Vectors and the Tail Point of the Resultant Vector is the Common Tail Point of the 2 Vectors.
   
   From Fig.3 as per Parallelogram Law of Vectors
   
   \(\vec{DB}=\vec{AB} + \vec{CB} = \vec{CB} + \vec{AB}\)
   
   And as per Cosine Law of Triangles
   
   \(|\vec{DB}|=|\vec{BD}|=\sqrt{|\vec{AB}|^2 + |\vec{CB}|^2 + 2|\vec{AB}||\vec{CB}|\cos \theta}\)
   
   Parallelogram Law can be used to Find the Resultant Vector under following conditions
   1. Addition of 2 Vectors that Have a Common Tail Point: The resultant Vector of addition of 2 Given Vectors \(\vec{AB}\) and \(\vec{CB}\) is given as
      
      \(\vec{AB} + \vec{CB} = \vec{CB} + \vec{AB}= \vec{DB}\)
   2. Addition of 2 Vectors that Have a Common Head Point: The resultant Vector of addition of 2 Given Vectors \(\vec{BA}\) and \(\vec{BC}\) is given as
      
      \(\vec{BA} + \vec{BC} = -(\vec{AB} + \vec{CB}) = -\vec{DB}=\vec{BD}\)
   3. Subtraction of 2 Vectors that are in Continuation with each other: The resultant Vector of Subtration of 2 Vectors \(\vec{AB}\) and \(\vec{BC}\) is given as
      
      \(\vec{AB} - \vec{BC} = \vec{AB} + \vec{CB} = \vec{DB}\)
      
      \(\vec{BC} - \vec{AB} = \vec{BC} + \vec{BA} = \vec{BA} + \vec{BC} = -(\vec{AB} + \vec{CB}) = -\vec{DB}=\vec{BD}\)