Let \(\vec{a}=2\overset{^}{i}+3\overset{^}{j}+4\overset{^}{k},\text{ }b=\overset{^}{i}−2\overset{^}{j}−2\overset{^}{k}\)…

Let \(\vec{a}=2\overset{^}{i}+3\overset{^}{j}+4\overset{^}{k},\text{  }b=\overset{^}{i}−2\overset{^}{j}−2\overset{^}{k}\) and \(\vec{c}=−\overset{^}{i}+4\overset{^}{j}+3\overset{^}{k}.\) if \(\vec{d}\) is a vector perpendicular to both \(\vec{b}\text{ and  }\vec{c}\), and \(\vec{a}.\vec{d}=18,\) then \({\left|\vec{a}\times \vec{d}\right|}^{2}\) is equal to:

[JEE Main 2023, 6 Apr (Shift 1)]