Definition of a vector using transformations

[beeftrax]

In the "intro to differential forms" thread by lethe, Super Mentor Tom defines a vector as something that transforms under rotation (multiplication by an orthogonal matrix) and parity (reflection through a mirror) in a certain way. I'm currently reading "Introduction to Vector and Tensor Analysis" by Robert Wrede, which uses the transformations of rotation and translation instead. So I have two questions:

1). Why explicitly define how a vector must change in a parity transformation? Isn't this just a special case of rotation, with the angle being 180?

2). Are Tom and Wrede actually defining two different types of vectors, or defining the same thing in two different ways?

 

[lethe]

Originally posted by beeftrax

1). Why explicitly define how a vector must change in a parity transformation? Isn't this just a special case of rotation, with the angle being 180?

no. for example, rotation by 180 degrees around the z-axis in R³ takes the x-axis to negative x-axis, and likewise the y-axis, but it leaves the z-axis fixed. a parity transformation exchanges all 3 axes, and cannot be achieved by any rotation.

2). Are Tom and Wrede actually defining two different types of vectors, or defining the same thing in two different ways?

so Tom wants a vector to be something that lives in the vector representation of SO(3), and Wrede wants it to be something that lives in the vector representation of ISO(3) (inhomogeneous SO(3)), and you want to know if those are different vectors.

since the SO(3) group is a subgroup of ISO(3), i think we can regard these vectors as the same, i.e. living in the same vector space.

 

[beeftrax]

Thank you very much for replying. I have a few more questions, if you don't mind.

1). What is ISO(3) exactly? Is it a combination of the rotation group and translation group for 3-space?

2). Is Wrede mistaken in not referring to parity in his definition of vector, or what?

 

[lethe]

Originally posted by beeftrax
Thank you very much for replying. I have a few more questions, if you don't mind.

1). What is ISO(3) exactly? Is it a combination of the rotation group and translation group for 3-space?

that is exactly what ISO(3) is.

2). Is Wrede mistaken in not referring to parity in his definition of vector, or what?

not mistaken. its just a matter of taste. vectors that transform nontrivially under parity are sometimes called pseudovectors, but if Wrede doesn t care so much about parity, its nothing to get upset over. its not always important.

 

[DivisionByZero]

What exactly is an orthogonal matrix anyway? I know one thing often used in graphics programing often times called a transformation matrix which Is actually 4x4 but can be 3x3 as the other numbers can be assumed. the matrix can be multiplied(I think) with a vector to transfer it from local to Global space. division transfers from global to local I think. Really, all it is is three vectors representing the three axis of a local space. Hopefully you can just say 'yes, that's an orthogonal matrix' but if not, what is it?

Also, what is an inertial tensor(or whatever it is)? I've been working on using this very nice physics system called tokamak. (software), and you have to set the inertial tensors for an object. Most of the time we just leave it at .003 or somthing and it works fine, however, I would like to know what the real values are. The only possible physical objects are a capsule(cylinder with a semisphere on both ends), a box, and a sphere. Are their some equations (without delving into calculus) that you can just plug in some values related to the object and get this 'mystical' :) inertial tensor?