The matrix representation of \(\mathrm{C}_{3 \mathrm{v}}\) given in Eq. (5.14) was constructed with respect to the particular coordinate system defined by the unit vectors \(\boldsymbol{e}_{1}\) and \(\boldsymbol{e}_{2}\) in Fig. 5.10 . Show that a corresponding matrix representation for basis vectors rotated clockwise by \(\frac{\pi}{6}\) relative to those in Fig. 5.10 is given by Eq. (5.15). Hint: We could proceed geometrically as in Fig. 5.10 , but a more elegant approach is to exploit rotational symmetry and use the rotation operator of Eq. (6.3) to perform a similarity transformation (2.12) on the original matrices.