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Difference between line vector and free vector

  1. Nov 26, 2013 #1
    Hi,

    I started to study Roy Featherstone's book "Rigid Body Dynamics Algorythms".

    The book starts off by explaining Spatial Algebra, where translations and rotations are gathered in a 6-D vector using Plucker's coordinates.

    At some point the book says;

    "A line vector is a quantity that is characterized by a directed line and a magnitude.
    A pure rotation of a rigid body is a line vector, and so is a linear force
    acting on a rigid body. A free vector is a quantity that can be characterized by
    a magnitude and a direction. Pure translations of a rigid body are free vectors,
    and so are pure couples. A line vector can be specified by five numbers, and
    a free vector by three. A line vector can also be specified by a free vector and
    any one point on the line."

    Can someone explain the difference between line vector and free vector in different words, especially the part where a line vector can be specified by five numbers.

    my current understanding would be that a free vector is the common euclidean vector, but then in the formulation above the two seem to differ by the fact that one is characterized by a directed line and the other by a direction. What's the difference ?

    Disclaimer; most of my algebra is self taught so if say some non-sense, that's why :)

    Hopefully this is the right location for such a post.

    Michael
     
  2. jcsd
  3. Nov 26, 2013 #2

    jedishrfu

    Staff: Mentor

    Welcome to PF!

    My understanding is a free vector has direction and magnitude.

    and it sounds like a line vector has additional components for theta and phi rotational angles (think spherical coordinate angles).

    So a line vector could be used to describe a physical object in space including its rotation about the vector.

    Does that make sense?

    This may help too:

    http://en.wikipedia.org/wiki/Rigid_body_dynamics

    http://en.wikipedia.org/wiki/Rigid_body
     
    Last edited: Nov 26, 2013
  4. Nov 27, 2013 #3
    Thank you, it makes sense for the most part :)

    so if a line vector is defined by a vector plus theta and phi (relative to the basis axis?) the text would mean;

    "A pure rotation of a rigid body is a line vector"; the vector would be 0 and theta and phi would be the rotation of the body relatively to the basis of the space.

    "so is a linear force acting on a rigid body"; that one i don't get, why would a linear force need to be a line vector and not just a free vector.

    "Pure translations of a rigid body are free vectors"; Ok

    "so are pure couples"; OK but then how is it different from the linear force?
     
  5. Nov 27, 2013 #4
    Here is the following part of the text that may help too, it's quite confusing to me.

    "Let ^s be any spatial vector, motion or force, and let s and sO be the two 3D coordinate vectors that supply the Plucker coordinates of ^s."
    [me; in Plucker coordinates s are 3 rotations angles around the basis axis, and sO a vector, though here it seems to be saying that they are 2 vectors]

    "Here are some basic facts about line vectors and free vectors.
     If s = 0 then ^s is a free vector."
    [me: yes only sO is left and that's is a vector]

    "If s.sO = 0 then ^s is a line vector. The direction of the line is given by
    s, and the line itself is the set of points P that satisfy OP X s = sO."
    [me: in order to do a dot product here, it has to be 2 vectors, but then that's different from Plucker coordinates?]
    
    "Any spatial vector can be expressed as the sum of a line vector and a free vector. If the line vector must pass through a given point, then the expression is unique.
    
    Any spatial vector, other than a free vector, can be expressed uniquely as
    the sum of a line vector and a parallel free vector. The expression (for a
    motion vector) is
    
    Code (Text):
    |     s    |     |  0   |
    | sO - hs  |   + |  hs  | where h =  (s.sO) / (s.s)
    [me: The | above are supposed to be big [] ]

    This last result implies that any spatial vector, other than a free vector, can be described uniquely by a directed line, a linear magnitude and an angular magnitude. Free vectors can also be described in this manner, but the description is not unique, as only the direction of the line matters. "

    Any help with this would be much appreciated.
     
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