Find a unit vector in the same direction as the given vector \( \vec{v} = 3\hat{i} - 4\hat{j} + 2\hat{k} \).

Step 1 ：Step 1: Calculate the magnitude of the given vector. The magnitude of a vector \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by \( \sqrt{a^2 + b^2 + c^2} \). So, the magnitude of \( \vec{v} \) is \( \sqrt{3^2 + (-4)^2 + 2^2} = \sqrt{9 + 16 + 4} = \sqrt{29} \).

Step 2 ：Step 2: Determine the unit vector in the same direction as \( \vec{v} \). A unit vector in the same direction as a given vector \( \vec{v} \) is given by \( \frac{\vec{v}}{||\vec{v}||} \). So, the unit vector in the same direction as \( \vec{v} \) is \( \frac{3\hat{i} - 4\hat{j} + 2\hat{k}}{\sqrt{29}} = \frac{3}{\sqrt{29}}\hat{i} - \frac{4}{\sqrt{29}}\hat{j} + \frac{2}{\sqrt{29}}\hat{k} \).