In physics, an orthonormal basis is often preferred over an arbitrary basis, such as a Cartesian basis, in three-dimensional space for several reasons:
Simplicity and Convenience: An orthonormal basis simplifies calculations and mathematical expressions. The components of vectors and tensors become easier to handle when the basis vectors are orthogonal (perpendicular to each other) and normalized (unit length). This simplification allows for straightforward manipulation of vector equations and facilitates mathematical operations like dot products and cross products.
Preservation of Geometric Relationships: Orthonormal bases preserve geometric relationships, such as angles and distances. When working with an orthonormal basis, the dot product of two vectors yields the cosine of the angle between them, which makes it easier to reason about angles and projections. It also enables us to relate vector magnitudes to lengths and distances in a more intuitive manner.
Consistency with Physical Laws: Many physical laws and equations are derived and expressed using an orthonormal basis. For example, Maxwell's equations, which describe electromagnetic phenomena, are often formulated using Cartesian coordinates, where the electric and magnetic fields are expressed in terms of three orthogonal components.
Compatibility with Symmetry: Orthonormal bases align well with symmetry considerations in physics. Symmetries play a crucial role in understanding physical systems, and an orthonormal basis facilitates the identification and analysis of symmetry properties. Certain physical phenomena exhibit symmetries that align with the coordinate axes of an orthonormal basis, making it a natural choice for studying such systems.
While an arbitrary basis can be used in some cases, particularly when dealing with specific coordinate transformations or when the problem naturally lends itself to a non-orthonormal basis, the orthonormal basis is often the default choice in physics due to its simplicity, consistency, and compatibility with many physical principles and equations.