Step-by-step Solution:

1. Given Information: - The vector \(\vec{a}\) makes equal angles with the coordinate axes. - The magnitude of \(\vec{a}\) is 3. 2. Direction Cosines: Let the angles that \(\vec{a}\) makes with the \(x\)-axis, \(y\)-axis, and \(z\)-axis be \(\alpha\), \(\beta\), and \(\gamma\) respectively. Since the angles are equal: \[ \alpha = \beta = \gamma = \theta \] The direction cosines are: \[ \cos \alpha = \cos \beta = \cos \gamma = \cos \theta \] 3. Sum of Squares of Direction Cosines: The sum of the squares of the direction cosines is 1: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Since \(\alpha = \beta = \gamma = \theta\): \[ 3 \cos^2 \theta = 1 \Rightarrow \cos^2 \theta = \frac{1}{3} \Rightarrow \cos \theta = \frac{1}{\sqrt{3}} \] 4. Determine the Angle \(\theta\): The angle \(\theta\) is: \[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \] Therefore, the angle between \(\vec{a}\) and each of the three coordinate axes is: \[ \boxed{\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)} \]