Answer
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Step 1: Recall the definition of the dot product.
\vec{a} \cdot \vec{b} = a\_1b\_1 + a\_2b\_2 + a\_3b\_3
Step 2: Identify the components of the vectors given in the problem.
\vec{b} = 3\vec{i} + 2\vec{j} + \vec{k}
We have: and
Final Answer
\boxed{\vec{a} \cdot \vec{b} = 19}.
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