Do Pseudovectors Change Direction After Reflection?

[SgrA*]

The Wikipedia article on http://en.wikipedia.org/wiki/Pseudovector" initally suggests that they do change direction after reflection.

Geometrically it (a pseudovector) is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true or polar vector (more formally, a contravariant vector), which on reflection matches its mirror image.

But in a later http://en.wikipedia.org/wiki/Pseudovector#Physical_examples" angular momentum remains invariant under reflection.

Each wheel of a car driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car.

A pseudovector is direction invariant under reflection or not?

I tried to devise a trick on the Right Hand Rule - if I were to switch to a left hand rule (left hand appears to be a mirror image of the right hand), and curl my left palm as in right hand rule. I noticed that if the left hand is to be maintained as a mirror image of the right, the direction of both the thumbs is same, suggesting that it is invariant.

 

[D H]

The Wikipedia article on http://en.wikipedia.org/wiki/Pseudovector" initally suggests that they do change direction after reflection.

Geometrically it (a pseudovector) is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true or polar vector (more formally, a contravariant vector), which on reflection matches its mirror image.

But in a later http://en.wikipedia.org/wiki/Pseudovector#Physical_examples" angular momentum remains invariant under reflection.

Well, that's wikipedia for ya'. Sometimes very good, sometimes poorly written and confusing, sometimes flat out wrong. This article falls in the second category.

You are talking about this image in the wiki article:

Think of the car on the left as the real car, the car on the right as the car's reflection. Imagine a vector on the car's rear bumper, with the tail in the middle of the red/gray hatched region and the head in the middle of the red/blue hatched region. In the car on the left, this vector is parallel to the angular momentum vectors of the car's wheels. Upon reflection, this vector reverses direction: It is points to the right upon reflection. This displacement vector is a real vector rather than a pseudovector. It's reflected image is the reverse of the real image in this particular example. The angular momentum pseudovectors are reflected and reversed. Reflection alone would reverse the direction of the pseudovectors. Reversing the reflections of these pseudovectors leaves those pseudovectors unchanged in this particular example.

 

[Chandra Prayaga]

Actually, the original statement is not complete and not correct. It is true that a pseudovector, reflected in a mirror perpendicular to the vector, does not change sign. If you reflect a pseudovector in a mirror parallel to the vector, it does change sign.

 

[D H]

You misread that statement in the wikipedia article. The statement is correct, just very poorly worded.

 

[PJC01]

I would like to clarify that the direction of a pseudovector does not change after reflection. This is because a pseudovector is defined as a quantity that has magnitude and direction, but it also has an inherent sense of "handedness" or chirality. This handedness cannot be changed by a simple reflection, unlike a polar vector which can be reversed in direction.

In the examples given, such as the angular momentum of a car's wheels, the direction of the pseudovector remains the same even after reflection. This is because the sense of rotation or spin is maintained in both the original and reflected object.

In regards to the trick with the left hand rule, it is important to note that the left hand rule is used for polar vectors, not pseudovectors. The left hand rule can be used to determine the direction of the magnetic field around a current-carrying wire, for example. But for a pseudovector such as angular momentum, the right hand rule is used.

In summary, the direction of a pseudovector is invariant under reflection, as it is determined by its inherent handedness. This is an important concept to understand in physics, as it helps to differentiate between polar and pseudovectors and their properties.