Step-by-step Solution:

The direction cosines of a vector \(\vec{a} = (-2\hat{i} + \hat{j} - 5\hat{k})\) are given by the normalized components of the vector. The direction cosines are the cosines of the angles that the vector makes with the \(x\)-axis, \(y\)-axis, and \(z\)-axis. To find them, we need to first find the magnitude of the vector and then divide each component of the vector by its magnitude. The magnitude of \(\vec{a}\) is: \[ |\vec{a}| = \sqrt{(-2)^2 + 1^2 + (-5)^2} = \sqrt{4 + 1 + 25} = \sqrt{30} \] Now, we can find the direction cosines by dividing each component of \(\vec{a}\) by \(\sqrt{30}\): \[ \text{Direction cosines} = \left( \frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}} \right) \] Therefore, the correct answer is: \[ \left( \frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}} \right) \]