Let \(\vec{a}=4\overset{^}{i}+5\overset{^}{j}−\overset{^}{k}\), \(\vec{b}=\overset{^}{i}−4\overset{^}{j}+5\overset{^}{k}…

Let \(\vec{a}=4\overset{^}{i}+5\overset{^}{j}−\overset{^}{k}\), \(\vec{b}=\overset{^}{i}−4\overset{^}{j}+5\overset{^}{k}\) and \(\vec{c}=3\overset{^}{i}+\overset{^}{j}−\overset{^}{k}.\) Let \(\vec{d}\) be a vector which is perpendicular to both \(\vec{a}\) and \(\vec{b}\) and \(\vec{c}⋅\vec{d}=12.\) Then \((-\overset{^}{\mathrm{i}}+\overset{^}{\mathrm{j}}-\overset{^}{\mathrm{k}}).(\vec{\mathrm{c}}\times \vec{\mathrm{d}})\) is equal to

[JEE Main 2023, 10 Apr (Shift 2)]