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A '''vector space''' is one of the most fundamental and important concepts in mathematics, physics, and engineering. It has its origins in the notion of a "direction" and a "magnitude". Perhaps the simplest vector to visualize is the ''velocity vector'', showing the speed and direction of motion of a particle.  Other extremely common vectors are the [[electric field]] and [[magnetic field]] vectors, though vectors abound in numerous areas of mathematics, physics, mechanical engineering, and aspects of electrical engineering.

A '''vector space''' is a [[set]] of objects that can be added together and multiplied by elements of another set, while satisfying certain properties. Elements of the first set are called "vectors" while elements of the second set are called "scalars". The idea of a vector space is one of the most fundamental and important concepts in mathematics, physics, and engineering.  

 

In two dimensions, a vector has a "magnitude" (or length) and a "direction" (or angle).Perhaps the simplest vector to visualize is the ''velocity vector'', showing the speed and direction of motion of a particle.  Other extremely common vectors are the [[electric field]] and [[magnetic field]] vectors, though vectors abound in numerous areas of mathematics, physics, mechanical engineering, and aspects of electrical engineering.

The most important operations involving vectors are the ''vector sum'' and the ''vector-scalar product''.  As an example of the first, if we are in a train traveling with speed given by one vector, and we throw something inside the train with a velocity, relative to the train, of another vector, the velocity of the object relative to a fixed observer is the sum of those two vectors.  As an example of the second, if we double the current through an electromagnet, its magnetic field vector will be multiplied by the number 2.  That is, its direction will be unchanged and its magnitude will double.

The most important operations involving vectors are the ''vector sum'' and the ''vector-scalar product''.  As an example of the first, if we are in a train traveling with speed given by one vector, and we throw something inside the train with a velocity, relative to the train, of another vector, the velocity of the object relative to a fixed observer is the sum of those two vectors.  As an example of the second, if we double the current through an electromagnet, its magnetic field vector will be multiplied by the number 2.  That is, its direction will be unchanged and its magnitude will double.