Let \(\vec{\mathrm{a}}=2\overset{^}{\mathrm{i}}-3\overset{^}{\mathrm{j}}+\overset{^}{\mathrm{k}},\vec{\mathrm{b}}=3\over…

Let \(\vec{\mathrm{a}}=2\overset{^}{\mathrm{i}}-3\overset{^}{\mathrm{j}}+\overset{^}{\mathrm{k}},\vec{\mathrm{b}}=3\overset{^}{\mathrm{i}}+2\overset{^}{\mathrm{j}}+5\overset{^}{\mathrm{k}}\) and a vector \(\vec{c}\) be such that \((\vec{\mathrm{a}}-\vec{\mathrm{c}})\times \vec{\mathrm{b}}=-18\overset{^}{\mathrm{i}}-3\overset{^}{\mathrm{j}}+12\overset{^}{\mathrm{k}}\) and \(\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=3\). If \(\vec{\mathrm{b}}\times \vec{\mathrm{c}}=\vec{\mathrm{d}}\), then \(|\vec{\mathrm{a}}\cdot \vec{\mathrm{d}}|\) is equal to :

[JEE Main 2025, 2 Apr (Shift 2)]