Vector - General (Physical), Spatial Vector, Displacement Vector, Position Vector, Basis Vector,  Mathematical Coordinate Representation

General (Physical)- Representation of a magnitude and direction. Representation may be graphical and/or mathematical. Anything that has a magnitude and a direction can be represented as a vector. Examples include displacement from a point, position (see below), velocity, force, electric field, magnetic field, thermal gradient, acceleration, etc. Compare descriptor.

Spatial Vector - A vector denoting the spatial relationship (distance and direction). It may be used to represent  the relation between two points in space or the displacement of a point in space or any other distance-direction concept . See also translation vector.

Given two points A and B, is the vector with its base at A and tip at B. Its magnitude is the distance between A and B and its direction is in the direction   from A toward B.  Equivalent  vectors describe the same relationship between different sets of points. That is, if there is a point D which is  separated by the same distance and direction from C as  B  is from A, then is equivalent to (the same as) the vector.

Displacement Vector - A   vector denoting the changing spatial relationship (distance and direction) of a point (real or virtual, physical or mathematical). See also translation vector.

Position Vector - The spatial vector to a given point from (relative to) another reference point such as the origin of a coordinate system.

Basis Vector - The spatial vectors that are used by a   coordinate system (cs) to describe all other spatial vectors in that system using coordinates.

Basis vectors  (usually referred to as i,j, and k) can have any dimensions such as millimeters or inches. The direction and distance from the origin (position vector) for an object (at point P)can be determined by a linear combination of i,j, and k (ai+b,j+ck), The coordinates for P  would be (a, b, c)

Coordinate Representation of a Vector

• Mathematically, vectors are often represented as a list of  coordinates in a  reference frame.   For example,  = (3,4,5). These coordinates may represent a displacement vector, position vector, translation vector etc.The coordinates must be used with an understanding of what they represent and in what reference frame. See basis vector.
  
  
  • For a rectilinear coordinate system, if the units are understood or are included in an associated basis , the form is (number 1, number 2, number 3, ...). The magnitude = the square root of (the sum of the squares of each number) provided all numbers have same units.
    

   

• Corresponding displacement vector coordinates can be added or subtracted provided that the coordinate representations are from the same or equivalent (different origins if position isn't involved) reference frames. Displacement example:
  

• A vector can be formed using equations to replace each number. The equations can be a function of position coordinates or other variables. For example, the electric field within an object may be expressed in the form (Ax, By, Cz) where A, B, and C are constant coefficients which relate position coordinates x, y, z to electric field strength and direction.
  

• See also tutorial:
  Reference Coordinate Systems and Transformations