1. The problem is to simplify the given vector expressions for \( \vec{PQ} \) and express them in simplest form. 2. We will analyze each option and simplify the coefficients where possible by factoring and reducing fractions. 3. For option A: \( \vec{PQ} = \frac{3}{7} \vec{b} - \frac{4}{7} \vec{c} \). This is already in simplest form. 4. For option B: \( \vec{PQ} = -\frac{1}{14} \vec{b} - \frac{3}{7} \vec{c} \). The fractions cannot be simplified further. 5. For option C: \( \vec{PQ} = \frac{7}{2} \vec{b} - 4 \vec{c} \). This is already simplified. 6. For option D: \( \vec{PQ} = \frac{1}{14} \vec{b} - \frac{4}{7} \vec{c} \). Fractions are in simplest form. 7. For option E: \( \vec{PQ} = \frac{4}{7} \vec{b} - \frac{3}{7} \vec{c} \). This is simplest form. Since the problem does not specify further operations, the expressions are already simplified as given. Final answer: Each vector expression is already in simplest form as shown.