Vector Transformation Law

The set of $n$ quantities $v_j$ are components of an $n$-D Vector ${\bf v}$ Iff, under Rotation,

$$v_i'=a_{ij}v_j$$

for $i=1$, 2, ..., $n$. The Direction Cosines between $x_i'$ and $x_j$ are

$$a_{ij} \equiv {\partial x_i'\over\partial x_j} = {\partial x_j\over\partial x_i'}.$$

They satisfy the orthogonality condition

$$a_{ij}a_{ik} = {\partial x_j\over\partial x_i'}{\partial x_i... ...\partial x_k} ={\partial x_j\over\partial x_k} = \delta_{jk},$$

where $\delta_{jk}$ is the Kronecker Delta.

See also Tensor, Vector