1. **State the problem:** We need to divide the vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ into three components, where one component is parallel to the vector $\vec{i}$. 2. **Formula and rules:** The component of a vector $\vec{v}$ parallel to a unit vector $\vec{u}$ is given by the projection formula: $$\text{proj}_{\vec{u}} \vec{v} = \left( \frac{\vec{v} \cdot \vec{u}}{\|\vec{u}\|^2} \right) \vec{u}$$ Since $\vec{i}$ is a unit vector, $\|\vec{i}\| = 1$. 3. **Calculate the component parallel to $\vec{i}$:** $$\vec{v} \cdot \vec{i} = 3 \times 1 + 5 \times 0 + 1 \times 0 = 3$$ So, $$\text{proj}_{\vec{i}} \vec{v} = 3 \vec{i}$$ 4. **Find the remaining components:** The vector $\vec{v}$ can be written as the sum of three components: - Parallel to $\vec{i}$: $3\vec{i}$ - Parallel to $\vec{j}$: $a\vec{j}$ - Parallel to $\vec{k}$: $b\vec{k}$ Since $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$, the other two components are simply: $$5\vec{j} \quad \text{and} \quad \vec{k}$$ 5. **Final answer:** The three components of $\vec{v}$ are: $$3\vec{i}, \quad 5\vec{j}, \quad \vec{k}$$