1. **Problem (c):** Given vectors $\vec{QP} = 5\mathbf{i} - 3\mathbf{j}$ and $\vec{OQ} = 3\mathbf{i} + 5\mathbf{j}$, find the magnitude of $\vec{PQ}$. 2. **Recall:** The vector $\vec{PQ} = \vec{Q} - \vec{P}$. Since $\vec{QP} = \vec{P} - \vec{Q}$, we have $\vec{PQ} = -\vec{QP}$. 3. **Calculate $\vec{PQ}$:** $$\vec{PQ} = -\vec{QP} = -(5\mathbf{i} - 3\mathbf{j}) = -5\mathbf{i} + 3\mathbf{j}$$ 4. **Magnitude formula:** The magnitude of a vector $\vec{v} = a\mathbf{i} + b\mathbf{j}$ is $$|\vec{v}| = \sqrt{a^2 + b^2}$$ 5. **Calculate magnitude of $\vec{PQ}$:** $$|\vec{PQ}| = \sqrt{(-5)^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$$ 6. **Final answer:** The magnitude of $\vec{PQ}$ is $$\boxed{\sqrt{34}}$$