Let \(\vec{a}=4\hat{i}+5\hat{j}−\hat{k}\), \(\vec{b}=\hat{i}−4\hat{j}+5\hat{k}\) and \(\vec{c}=3\hat{i}+\hat{j}−\hat{k}.…

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

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