Since \(\vec{u}\) is coplanar with \(\vec{a}\) and \(\vec{b}\), we can write \(\vec{u} = x\vec{a} + y\vec{b} = x(2\hat{i}+3\hat{j}-\hat{k}) + y(\hat{j}+\hat{k}) = 2x\hat{i} + (3x+y)\hat{j} + (-x+y)\hat{k}\). Condition \(\vec{u} \perp \vec{a}\): \(\vec{u} \cdot \vec{a} = 2x(2) + (3x+y)(3) + (-x+y)(-1) = 0\), giving \(4x + 9x + 3y + x - y = 0\), so \(14x + 2y = 0\), thus \(y = -7x\). Condition \(\vec{u} \cdot \vec{b} = 24\): \((3x+y)(1) + (-x+y)(1) = 24\), giving \(3x+y-x+y = 24\), so \(2x+2y = 24\), thus \(x+y = 12\). From \(y=-7x\): \(x-7x = 12\), so \(-6x=12\) and \(x=-2\), giving \(y=14\). Thus \(\vec{u} = -4\hat{i} + 8\hat{j} + 16\hat{k}\), and \(|\vec{u}|^2 = 16 + 64 + 256 = 336\).