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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
Step 3:: Apply the formula for the dot product using the components of the vectors.
\vec{a} \cdot \vec{b} = (2)(3) + (4)(2) + (5)(1)
Step 4:: Calculate the result.
\vec{a} \cdot \vec{b} = 6 + 8 + 5 = 19
Final Answer
\boxed{\vec{a} \cdot \vec{b} = 19}.
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