Show that the vector product 𝐚 ×𝐛 of two classical vectors transforms like a vector under rotati

step 1 transform each vector and then multiply. That is the transformation law for a mixed second rank.

Show that the vector product 𝐚 ×𝐛 of two classical vectors...

Show that the vector product $\mathbf{a} \times \mathbf{b}$ of two classical vectors transforms like a vector under rotations. Hint: A rotation matrix $\mathbf{R}$ satisfies the relations $\mathbf{R} \cdot \mathbf{R}^{T}=\mathbf{I}$ and $\operatorname{det}(\mathbf{R})=1$, which in tensor notation read $\sum_{p} R_{i p} R_{t p}=\delta_{i t}$ and $\sum_{i j k} \epsilon_{i j k} R_{i r} R_{j s} R_{k t}=\epsilon_{r s t}$

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Key Concepts

Rotation Matrix

A rotation matrix is a square matrix that performs a rotation in Euclidean space. It has the property of orthogonality, meaning that the matrix times its transpose yields the identity matrix, and its determinant is equal to one. These properties ensure that the rotation preserves the length of vectors and the angles between them.

Vector Transformation

Under rotations, vectors transform by applying the rotation matrix to their components. This property guarantees that the geometric nature of vectors remains invariant under coordinate changes corresponding to rotations.

Cross Product and Its Transformation

The cross product of two vectors, defined through the antisymmetric Levi-Civita symbol, transforms like a vector under rotations. This means that if the original vectors are rotated using a rotation matrix, their cross product is also rotated by the same matrix, preserving its vector nature.

Levi-Civita Symbol

The Levi-Civita symbol is a completely antisymmetric tensor used to define the cross product in three dimensions. Under proper rotations, it transforms in such a way that the orientation and antisymmetric properties of the cross product are maintained, reinforcing the consistency of vector operations under coordinate transformations.

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