Introduction to Vector Algebra

1. As given in the Introduction to Matrix, Vector and Tensor Algebra, a Vector is a Tensor having 2 or More Elements organised in a Single Row or a Single Column. Hence, Vectors can be Represented by Matrices in form of Column Matrices or Row Matrices.
2. The Elements of a Vector are called Vector Components. The Number of Components present in a Vector gives the Dimension of the Vector. For example, Vectors with \(2, 3, ..., N\) Components are called 2-Dimensional, 3-Dimensional, ..., N-Dimensional Vectors.
3. Each Vector Component can be a Numerical Value or Variable or Function. Vectors having Only Real Value/Variable/Function Components are called Real Vectors. Vectors having Atleast One Complex or Pure Imaginary Value/Variable/Function Component are called Complex Vectors.
4. Vectors provide an Abstract Method for Representing Physical Quantities/Entities. A Vector can have as many Components as is required for representation of a particular Physical Quantity/Entity.
5. Every Vector has a Positive Real Valued Numerical Quantity (and a Complex Valued Numerical Quantity for Complex Vectors) associated with the Vector called the Magnitude or Norm or Length of the Vector. The Actual Meaning of the Magnitude/Norm/Length for a given Vector depends upon the Physical Quantity/Entity represented by the Vector.
6. Any Vector with a Magnitude/Norm/Length of Numerical Value 1 is called a Unit Vector.
7. Real Vectors are used for Representing Physical Quantities/Entities that have a Direction associated with them along with some Magnitude. The Magnitude of the Vector gives the Weightage or Impact of the Physical Quantity/Entity that the Vector has in the Direction of the Vector.
8. Components of any Real Vector are also called Direction Numbers of the Vector. The the Ratio between Direction Numbers of a Real Vector is called Direction Ratio.
9. The Components of any Real Unit Vector in the Direction of the Vector are called Direction Cosines of the Vector.
10. Complex Vectors are used for Representing Physical Quantities/Entities that have 2 or More States associated with them along with some Magnitude. The Actual Meaning of the Magnitude for a given Complex Vector depends upon the Physical Quantity/Entity represented by the Vector.