$\text{ If }\overrightarrow{a},\overrightarrow{b}\text{ are vectors perpendicular to each other and }\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=3,\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b},\text{ then the least value of }|\overrightarrow{c}-\overrightarrow{a}|\text{is}$

$\begin{array}{l}\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}\\ \Rightarrow |\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}|=\left|\overrightarrow{\mathrm{b}}\right|\\ \Rightarrow \left|\overrightarrow{\mathrm{c}}\right|\overrightarrow{\mathrm{a}}\mid \sin \mathrm{\theta }=3\\ \Rightarrow \left|\overrightarrow{\mathrm{c}}\right|=\frac{3}{2\sin \mathrm{\theta }}\\ \begin{array}{rl}|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}{|}^2& =|\overrightarrow{\mathrm{c}}{|}^2+|\overrightarrow{\mathrm{a}}{|}^2-2\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}}\\ & =|\overrightarrow{\mathrm{c}}{|}^2+4-2|\overrightarrow{\mathrm{c}}|\cdot |\overrightarrow{\mathrm{a}}|\cos \mathrm{\theta }\end{array}\end{array}$

$\begin{array}{l}=\frac{9}{4{\sin }^2\mathrm{\theta }}+4-2\cdot \frac{3}{2\sin \mathrm{\theta }}\cdot 2\cdot \cos \mathrm{\theta }\\ =4+\frac{9}{4}{\mathrm{cosec}}^2\mathrm{\theta }-6\cot \mathrm{\theta }\\ =\frac{9}{4}+{\left(\frac{3}{2}\cot \theta -2\right)}^2\\ =|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}{|}^2\ge \frac{9}{4}\Rightarrow |\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|\ge \frac{3}{2}\end{array}$