Step 1 :First, we find a vector that is orthogonal to both \(\vec{u} = \langle 3,4,-2 \rangle\) and \(\vec{v} = \langle 0,5,3 \rangle\) by taking their cross product. The cross product of \(\vec{u}\) and \(\vec{v}\) is \(\langle 22, -9, 15 \rangle\).
Step 2 :Next, we normalize this vector to get a unit vector. The normalized vector is \(\langle 0.78272487, -0.32020563, 0.53367605 \rangle\).
Step 3 :Then, we find a second vector that is orthogonal to both \(\vec{u}\) and \(\vec{v}\) by taking the cross product of the first unit vector we found and either \(\vec{u}\) or \(\vec{v}\). The cross product is \(\langle -1.49429294, 3.1664779, 4.09151639 \rangle\).
Step 4 :Finally, we normalize this vector to get the second unit vector. The normalized vector is \(\langle -0.27748323, 0.58800018, 0.75977552 \rangle\).
Step 5 :The two unit vectors orthogonal to both \(\vec{u} = \langle 3,4,-2 \rangle\) and \(\vec{v} = \langle 0,5,3 \rangle\) are \(\boxed{\langle 0.78272487, -0.32020563, 0.53367605 \rangle}\) and \(\boxed{\langle -0.27748323, 0.58800018, 0.75977552 \rangle}\).