1. **State the problem:** We need to divide the vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ into three components such that one component is parallel to the vector $\vec{i}$. 2. **Formula and explanation:** To find the component of $\vec{v}$ parallel to $\vec{i}$, we use the projection formula: $$\text{proj}_{\vec{i}} \vec{v} = \left( \frac{\vec{v} \cdot \vec{i}}{\|\vec{i}\|^2} \right) \vec{i}$$ Since $\vec{i}$ is a unit vector, $\|\vec{i}\| = 1$, so the formula simplifies to: $$\text{proj}_{\vec{i}} \vec{v} = (\vec{v} \cdot \vec{i}) \vec{i}$$ 3. **Calculate the dot product:** $$\vec{v} \cdot \vec{i} = 3 \times 1 + 5 \times 0 + 1 \times 0 = 3$$ 4. **Find the parallel component:** $$\vec{v}_{\parallel \vec{i}} = 3 \vec{i}$$ 5. **Find the perpendicular component to $\vec{i}$:** $$\vec{v}_{\perp \vec{i}} = \vec{v} - \vec{v}_{\parallel \vec{i}} = (3\vec{i} + 5\vec{j} + \vec{k}) - 3\vec{i} = 5\vec{j} + \vec{k}$$ 6. **Divide $\vec{v}_{\perp \vec{i}}$ into two components along $\vec{j}$ and $\vec{k}$:** $$\vec{v}_{\parallel \vec{j}} = 5\vec{j}$$ $$\vec{v}_{\parallel \vec{k}} = \vec{k}$$ 7. **Final components:** - Parallel to $\vec{i}$: $3\vec{i}$ - Parallel to $\vec{j}$: $5\vec{j}$ - Parallel to $\vec{k}$: $\vec{k}$ Thus, the vector $\vec{v}$ is decomposed as: $$\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$$ This matches the original vector, confirming the components are correct.