In physics, an orthonormal basis is often preferred in three-dimensional space because it offers several advantages and simplifications in mathematical and physical calculations. Here are some reasons why an orthonormal basis is commonly used:
Simplicity of vector representation: An orthonormal basis, such as the Cartesian coordinate system, provides a simple and intuitive way to represent vectors. Each vector can be uniquely expressed as a linear combination of basis vectors, with the coefficients representing the components of the vector. This simplifies vector operations such as addition, subtraction, and scalar multiplication.
Orthogonality simplifies calculations: In an orthonormal basis, the basis vectors are mutually orthogonal, meaning they are perpendicular to each other. This orthogonality property simplifies many calculations because the dot product of orthogonal vectors is zero. It allows for easy determination of the components of a vector along each basis direction and simplifies calculations involving vector projections and decompositions.
Preserving geometric properties: An orthonormal basis preserves important geometric properties. For example, in a Cartesian coordinate system, the length of a vector is given by the Pythagorean theorem, and angles between vectors can be easily calculated using dot products. These properties are crucial for analyzing physical quantities like forces, velocities, and accelerations.
Compatibility with physical laws and equations: Many fundamental physical laws and equations are formulated and expressed in terms of an orthonormal basis. For instance, Maxwell's equations in electromagnetism, Schrödinger's equation in quantum mechanics, and Newton's laws of motion in classical mechanics are often written in Cartesian coordinates. Using an orthonormal basis ensures compatibility with these equations and facilitates their application in various physical scenarios.
Experimental convenience: Orthonormal bases like Cartesian coordinates have been widely adopted in experimental setups and measurements. They provide a convenient framework for positioning and aligning instruments, determining distances and angles, and analyzing experimental data. Using an arbitrary basis can introduce additional complexities and make experimental measurements more challenging to interpret.
While an arbitrary basis could be used in specific cases or coordinate systems tailored for certain problems, the adoption of an orthonormal basis, particularly Cartesian coordinates, has become the standard in many branches of physics due to its simplicity, compatibility with equations, and preservation of geometric properties.