While the numerical components of a vector change when switching between different orthonormal bases, the vector itself (its physical direction and magnitude) remains invariant. This transformation between orthonormal bases is described by a rotation matrix, whose elements are the dot products of the old and new basis vectors. These transformation coefficients ensure that fundamental properties like vector magnitude and the scalar product (angle between vectors) are preserved, highlighting their role as invariants in physics.

Vectors are independent of the basis chosen to represent them, meaning the geometric or physical vector itself does not change when we change coordinate systems or bases. However, the components of that vector do change according to how the basis changes. This change of components for vectors under rotation is governed by rotation matrices.

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🎬Animated result and interactive web

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Here is a clear explanation:

• A vector $x$ is an abstract geometric object independent of basis. When expressed in a basis $\left\{ e _i\right\}$, it corresponds to components $x_i$, i.e.,

  $$ x =\sum_i x_i e _i . $$

• If we change the basis to $\left\{ e _i^{\prime}\right\}$, related to the old basis by a rotation matrix $R$, the vector itself remains the same but its components transform as:

  $$ x_i^{\prime}=\sum_j R_{i j} x_j, $$

where $R$ is a special orthonormal matrix representing the rotation (change of basis)

• The rotation matrix $R$ represents an orthogonal transformation preserving length and angles, satisfying $R^T=R^{-1}$. This means under rotation the magnitude of the vector and the dot products with other vectors remain unchanged.
• This process is sometimes called a passive rotation because it corresponds to changing the coordinate system or basis, not the vector itself. The vector is the same geometric object; only the coordinates used to describe it have changed.
• For tensors, which generalize vectors, the components transform similarly but with multiple rotation matrices (one per index). For a rank-1 tensor (vector), transformation is just by one rotation matrix; for higher-rank tensors, the transformation rule involves products of rotation matrices corresponding to each index.

This fundamental principle allows vectors and tensors to be treated consistently across different coordinate systems, ensuring physical laws are basis-independent even though their components depend on the chosen frame.

This section demonstrates rotations and basis changes in cloud computing, offering both an animated web visualization of 2D base rotations and vector transformations, alongside a numerical analysis of vector component changes in a new basis.

Vectors are Independent of Basis, Components Transform via Rotation Matrices

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1. Understanding Vectors and Their Operations
2. Applications and Visualization of Cross Product Orthogonality
3. Vectors are Independent of Basis, Components Transform via Rotation Matrices
4. The Kronecker Delta and Permutation Symbol are Essential Tools for Vector Algebra and Geometric Interpretation
5. Fields as Functions Mapping Space to Physical Quantities
6. Integral Theorems Connecting Derivatives to Boundaries
7. Vector Calculus in General and Orthogonal Coordinate Systems
8. Scalar and Vector Potentials: Decomposing Vector Fields and Their Properties

🫧Condensed notes-2

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🎬Animated result and interactive web

https://www.youtube.com/playlist?list=PLG2qYtL4WVLfewOyixUqf8c4HpK7LZn15

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1. Two possible bases in two dimensions related by a rotation with an angle
2. how a vector's components transform between two different orthonormal bases related by a rotation </aside>