Incariance, Consider a vector $\mathbf{A}$ in the cartesian coordinate system with unit vectors $\hat{x}, \hat{y}$, and $\hat{z}$. This system is now rotated through an angle $\theta$ about the $\hat{\mathbf{z}}$ axis.
(a) Express the new unit vectors $\hat{\mathbf{x}}^{\prime}$ and $\hat{y}^{\prime}$ in terms of $\hat{\mathbf{x}}, \hat{y}$ and $\theta ; \hat{\mathbf{z}}^{\prime}=\hat{\mathbf{z}}$
(b) Express $\mathbf{A}$ in terms of $A_{x^{\prime}}^{\prime}, A_{y^{\prime}}^{\prime}, A_{z^{\prime}}^{\prime}$ and $\hat{x}, \hat{y}^{\prime}, \hat{z} ;$ transform
to $\hat{\mathbf{x}}, \hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ and so find the relations between $A_{x^{\prime}}^{\prime}, A_{\dot{y}^{\prime}}$
$A_{z^{\prime}}^{\prime}$ and $A_{x^{\prime}} A_{y^{\prime}} A_{x^{2}}$
(c) Show that $A_{x}{ }^{2}+A_{y}{ }^{2}+A_{z}{ }^{2}=A_{x^{\prime}}^{\prime 2}+A_{v^{\prime}}^{\prime 2}+A_{z^{\prime}}^{\prime 2}$.
(This problem with an arbitrary rotation in three dimensions is complicated. One method is to use nine direction cosines among which there are six relations, three from the orthogonality of $\hat{x}^{\prime}, \hat{y}^{\prime}, \hat{z}^{\prime}$ and three from the fact that the sum of the squares of direction cosines is 1.)