Difference between line vector and free vector

[mytch]

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

 

[jedishrfu]

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

 

[mytch]

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?

 

[mytch]

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


|     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.

 

[blue_raver22]

Hi Michael,

Thank you for your question about the difference between line vector and free vector. I am happy to provide an explanation for you.

A line vector is a mathematical quantity that has both magnitude and direction, but is also characterized by a specific line. This means that in addition to the magnitude and direction, a line vector is also defined by the line it is associated with. In contrast, a free vector is a mathematical quantity that only has magnitude and direction, without any specific association to a line.

In the context of rigid body dynamics, a line vector is used to represent pure rotations, which involve movement along a specific line in space. On the other hand, a free vector is used to represent pure translations, which involve movement in a specific direction without any restriction to a specific line.

The statement in the book about a line vector being specified by five numbers refers to the Plucker coordinates, which are a way of representing a line in three-dimensional space using five numbers. These five numbers represent the direction and position of the line in space. In contrast, a free vector only requires three numbers to represent its magnitude and direction.

In simpler terms, a line vector is a more specific type of vector that is associated with a specific line, while a free vector is a more general type of vector that only has magnitude and direction. I hope this explanation helps clarify the difference between the two concepts for you.

Best,

 

[blue_raver22]

I can provide some clarification on the difference between line vectors and free vectors. In simple terms, a line vector represents a quantity that has both magnitude and direction, but is also associated with a specific line in space. This means that the vector's direction is defined by the line it is associated with, and the magnitude is the length of the vector along that line. On the other hand, a free vector has both magnitude and direction, but is not associated with any specific line in space. This means that the vector's direction can be defined independently from any specific line, and the magnitude is simply the length of the vector in that direction.

In terms of the numbers used to specify these vectors, a line vector requires five numbers because it needs to specify the line it is associated with, which can be done using three numbers (such as the coordinates of a point on the line) and two additional numbers to represent the direction along that line. A free vector only requires three numbers because it only needs to specify the direction and magnitude, which can be done using two numbers to represent the direction and one number for the magnitude.

To further clarify, think of a line vector as a vector that is "locked" onto a specific line in space, while a free vector is "free" to move and change direction without being attached to a specific line. This distinction is important in rigid body dynamics, where the motion of a rigid body can be described using a combination of line and free vectors. I hope this helps to better understand the difference between these two types of vectors.