Differences between an axial vector, a pseudo vector and a bivector?

In summary, there are three types of geometric objects: axial vectors, pseudo-vectors, and bivectors. Axial vectors and pseudo-vectors are dual to bivectors, with axial vectors representing directed lengths and pseudo-vectors representing directed areas. Bivectors can also be represented by numbers, known as pseudoscalars. A bivector is a directed area, similar to how a vector is a directed length, and in three dimensions it can be represented by its normal vector or "axis". The cross product of two vectors, represented as \vec{C} = \vec{A}\times\vec{B}, can be both a pseudo-vector and a bivector, where its magnitude is equal to the area
  • #1
Swapnil
459
6
What is the difference between an axial vector, a psudo vector and a bivector?
 
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  • #2
Axial vector and pseudo-vector seem to mean the same thing, and are dual to bivectors.

A bivector is a "directed area". (similar to how a vector is a "directed length") In three dimensions, a directed area can be represented by its normal vector (i.e. it's "axis"): that's where pseudovectors come from.

Similarly, in three dimensions, a "directed volume" can be represented by a number: that's where pseudoscalars come from.
 
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  • #3
So, if
[tex] \vec{C} = \vec{A}\times\vec{B}[/tex]

then C would be a psudo vector and it would also be a bivector since its magnitude is equal to the area of the parallogram spanned by A and B?
 
  • #4
Almost: [itex]A \times B[/itex] is the (pseudo)vector that is dual to the bivector [itex]A \wedge B[/itex].

[itex]A \times B[/itex] is merely the properly oriented vector that is perpendicular to the oriented parallelogram with sides A and B. Roughly speaking, [itex]A \wedge B[/itex] is that oriented parallelogram.

(But only roughly speaking -- the picture isn't quite that nice. For example, [itex](A + B) \wedge B = A \wedge B[/itex])
 

1. What is an axial vector?

An axial vector is a mathematical object that represents a quantity that has both magnitude and direction, but behaves differently under certain transformations. It is also known as a polar vector or a proper vector.

2. What is a pseudo vector?

A pseudo vector is a mathematical object that also has magnitude and direction, but its behavior is different from an axial vector under certain transformations. It is also known as an axial vector or an improper vector.

3. What is the difference between an axial vector and a pseudo vector?

The main difference between an axial vector and a pseudo vector is how they behave under certain transformations. An axial vector remains unchanged when the coordinate system is rotated, while a pseudo vector changes sign. Additionally, an axial vector follows the right-hand rule for cross products, while a pseudo vector follows the left-hand rule.

4. What is a bivector?

A bivector is a mathematical object that represents a directed area in three-dimensional space. It can be thought of as a combination of two vectors and has both magnitude and direction, but its behavior is different from both axial and pseudo vectors under transformations.

5. How are axial vectors, pseudo vectors, and bivectors used in physics?

These mathematical objects are frequently used in physics to represent physical quantities such as angular momentum, torque, and magnetic fields. They also play a crucial role in describing the behavior of particles and fields in quantum mechanics and relativity.

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