Pseudovector

In physics and mathematics, a pseudovector (or axial vector) is a special kind of quantity. It behaves like a regular vector in many ways, such as when an object is turned or moved smoothly. However, it acts differently when there is a mirror reflection or other sudden changes in direction.

One clear example of a pseudovector is the angular velocity of a spinning object. If you look at this object in a mirror, the direction of its spin does not match the mirror image as you might expect with a normal vector. This unusual behavior is important in many areas, such as computer graphics, where surface directions need to be handled correctly.

Pseudovectors appear in several important physics concepts, like the magnetic field and torque. In math, especially in three dimensions, pseudovectors are closely related to objects called bivectors. Understanding pseudovectors helps scientists and engineers solve problems in many different fields.

Physical examples

Physical examples of pseudovectors include angular velocity, angular acceleration, angular momentum, torque, magnetic field, and magnetic dipole moment.

When you look at something spinning, like the wheels of a car, they have a special kind of direction called angular momentum. If you were to look at the car in a mirror, the direction of this angular momentum would not look the same as in the real car. This is because angular momentum is a pseudovector, which behaves differently from regular vectors when you look in a mirror. This difference helps scientists understand how certain physical systems behave, especially when they are mirrored or flipped.

Details

See also: Covariance and contravariance of vectors and Euclidean vector

In physics, vectors are special kinds of quantities that change in a certain way when we rotate or reflect things. True vectors, called polar vectors, look the same in a mirror as they do in real life. But there are also special vectors called pseudovectors, or axial vectors, that change in a surprising way when we look at them in a mirror.

One easy example is the normal to an oriented plane. If you have a flat surface and you point your finger straight up from it, that direction is a pseudovector. If you look at that same surface in a mirror, your finger now points the opposite way compared to what you'd expect for a regular vector. This unusual behavior makes pseudovectors important in understanding rotations and reflections in physics.

The right-hand rule

When we talk about pseudovectors, we can think of switching from the "right-hand rule" to the "left-hand rule" in our math and physics. This means that while regular vectors stay the same, pseudovectors—like the magnetic field—would change their direction. Even with this switch, nothing physical would change, except in special cases like certain types of radioactive decays.

This idea helps us understand how pseudovectors behave differently from regular vectors when we look at things from different viewpoints.

Main article: right-hand rule
Main articles: cross product, curl
Further information: parity-violating, radioactive decays

Formalization

A pseudovector is a special kind of quantity in physics and math. Think of it like a direction that changes in a mirror differently than a normal direction. For example, if you spin a toy and look at it in a mirror, the spin direction in the mirror doesn’t match the original spin direction the same way a normal direction would.

There are a few ways to describe pseudovectors using advanced math, but the main idea is that they behave differently from regular vectors when you flip or mirror things. This makes them useful for describing things like spin or rotation in objects.

Main article: Pseudovector

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Geometric algebra

In geometric algebra, the basic elements are vectors. These vectors are used to build more complex elements using special rules for multiplying them together.

The main way to multiply vectors in this algebra is called the geometric product. When you multiply two vectors, a and b, you get two parts: one is the dot product, which tells you how much the vectors point in the same direction, and the other is called the wedge product or exterior product, which tells you about the area between the vectors.

A pseudovector is a special kind of object made from these wedge products. In three dimensions, a pseudovector is made by combining two vectors using the wedge product. This is different from regular vectors because it behaves specially when you look at things in a mirror or flip the directions of your space.