Noting the hierarchy between three mixing angles, ${\ensuremath{\theta}}_{2,3}=\mathcal{O}({\ensuremath{\theta}}_{1}^{2})$, we present an exact form of the quark mixing matrix, replacing Wolfenstein's approximate form. In addition, we suggest to rotate the unitarity triangle, using the weak $CP$ phase convention where the phase is located at the (31) element $\mathrm{sin}{\ensuremath{\theta}}_{1}\mathrm{sin}{\ensuremath{\theta}}_{2}{e}^{i\ensuremath{\delta}}$ while the (13) element $\mathrm{sin}{\ensuremath{\theta}}_{1}\mathrm{sin}{\ensuremath{\theta}}_{3}$ is real. For the $(ab)$ unitarity triangle, the base line ($x$-axis) is defined from the product of the first row elements, ${V}_{1a}{V}_{1b}^{*}$, and the angle between two sides at the origin is defined to be the phase $\ensuremath{\delta}$. This is a useful definition since every Jarlskog triangle has the angle $\ensuremath{\delta}$ at the origin, visible directly in the Cabibbo-Kobayashi-Maskawa matrix. It is argued that $\ensuremath{\delta}$ represents the barometer of the weak $CP$ violation, which can be related to possible Yukawa textures.