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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 and physics.  Not everything is a vector: an example of something that is not a vector is temperature, as it has no direction.  It is called a scalar rather than a vector.  But the ''change'' in temperature is a vector.

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 and physics.  Not everything is a vector: an example of something that is not a vector is temperature, as it has no direction.  It is called a "[[scalar]]" rather than a vector.  But the ''change'' in temperature is a vector.

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.