A velocity vector is first and foremost a vector. That is, it has both a magnitude and a direction. I will not discuss vectors here; that will be left up to the vector node. Suffice it to say that, in Cartesian coordinates, a velocity vector can be written as a set of components. A 3-dimensional velocity vector, v can be written in the form of <vx, vy, vz>.

The velocity vector is a vector which represents the velocity of an object. The velocity of an object is simply the distance travelled by the object per unit time. As stated, this is only a magnitude, but velocity is in reality directional in nature. More precisely stated, a velocity is the time derivative of the positional function of the object. That is, it is the change in the position of the object per unit time. Thus, if supplied a function which gives the position, r of an object at any given point in time, say <rx, ry, rz>, the velocity can be determined by:

  • vx = d/dt(rx)
  • vy = d/dt(ry)
  • vz = d/dt(rz)
More generally stated, vi = d/dt(ri), where ri and vi are the magnitude of r and v vectors, respectively, in the direction the i-axis. For example, for vx and rx, i = 1. For vy and ry, i = 2. Because of the componentwise nature of the vector derivative, the above can be written more simply as: v = d/dt(r).

Naturally, then, r can be determined by integrating each component of v. The acceleration vector, a is just the time derivative of v, and thus v can be found by integrating v. A quick overview:

  I |  acceleration    /\ 
  N |                   | 
  T |                   | D
  E |                   | E
  G |  velocity         | R
  R |                   | I
  A |                   | V
  T |                   | E
  E \/ position         |
Although not necessarily apparent, velocity is a vector, and thus, a velocity vector is simply a vector which stores a velocity. The velocity is not in the vector itself; it's in the use or interpretation of the vector.

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