Vector Types and their Diagramatic / Visual / Symbolic Representation

1. Vectors are Symbolically Represented using some Letter of an Alphabet. The Letter Symbols representing Vectors are often given in Boldface, or in case of Real Vectors with an Arrow on Top of them. For example any given Vector \(A\) can also be Symbolically Represented as \(\mathbf{A}\) (Boldface) or \(\vec{A}\) (with Arrow on Top if \(A\) is a Real Vector).
2. Given any Vector \(A\) / \(\mathbf{A}\) / \(\vec{A}\), the Symbol for Magnitude of the Vector \(A\) is given as \(|A|\) / \(|\mathbf{A}|\) / \(|\vec{A}|\).
3. Given any Vector \(A\) / \(\mathbf{A}\) / \(\vec{A}\), the Symbol for Unit Vector in the Direction of Vector \(A\) (or Corresponding to Vector \(A\)) is given as \(\hat{A}\) / \(\mathbf{\hat{A}}\).
4. Any 2 Vectors \(A\) and \(B\) are Equal only when the Corresponding Components of Both Vectors are Same, which implies \(\hat{A} = \hat{B}\) and \(|A| = |B|\) (i.e. Both their Magnitudes and Unit Vectors are Same).
5. Any 2 Vectors \(A\) and \(B\) are said to have the Same Direction only when \(\hat{A} = \hat{B}\) (i.e. only if Both their Unit Vectors are Same).
6. Any 2 Vectors \(A\) and \(B\) are said to have the Same Line Direction only when \(\hat{A} = \hat{B}\) or \(\hat{A} = \hat{-B}\) (i.e. only if Both their Unit Vectors are Same or are Negative of each other).
7. Real Vectors are Diagramatically represented using Line Segments with Arrow Heads in Cartesian Coordinate Systems as given in the following diagram
   
   
   The Direction of the Arrow Head gives the Direction of the Vector and is known as the Head of the Vector. The Opposite Direction is known as the Tail of the Vector.
8. Real Vectors can be of following 3 types
   1. Position Vectors: These Vectors have their Starting Points/Locations fixed at the Origin of the Cartesian Coordinate Systems. The Components of these Vectors give Absolute Locations/Positions in the Cartesian Coordinate Systems that correspond to the Ending Points/Locations of these Vectors. These Vectors are Labelled/Symbolized using an Alphabet either in Boldface or with an Arrow on Top of it or Both (as given for Vector \(\vec{A}\) in Fig. 1).
   2. Direction Vectors: These are also known as Floating Vectors or Free Vectors as they Do Not have their Starting or Ending Points/Locations fixed. The Components of these Vectors give the Relative Span/Extent of the Vector in the Direction of their Respective Basis Vectors. These Vectors are also Labelled/Symbolized using an Alphabet either in Boldface or with an Arrow on Top of it or Both (as given for Vector \(\vec{A}\) in Fig. 1). Real Unit Vectors, Direction Ratio Vectors of Lines, Normal and Tangent Vectors of Curves, Planes and Surfaces, Parallel and Perpendicular Components of a Vector (Projection and Rejection Vectors) are examples of Direction Vectors.
   3. Static Vectors: These Vectors have their Starting and Ending Points/Locations fixed. The Components of these Vectors are Difference Between the Corresponding Components of Ending and Starting Points/Locations in the Cartisean Coordinate Systems. These Vectors are Labelled/Symbolized using their Endpoint Lables, either in Boldface or with an Arrow on Top of them or Both (as given for Vector \(\vec{AB}\) & vector \(\vec{BA}\) in Fig. 2 & Fig. 3 respectively). The Point that is Mentioned First is the Direction or Head of the Vector and the Point that is Mentioned Second is the Tail of the Vector. So Fig. 2 is labelled as the vector \(\vec{AB}\) and Fig. 3 is labelled as the vector \(\vec{BA}\). Interchanging the Head and Tail of the Vector changes the Sign of the Vector. That is \(\vec{AB}=-\vec{BA}\) and \(\vec{BA}=-\vec{AB}\).