1. **State the problem:** We are given a vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ and need to divide it into three components: one parallel to $\vec{i}$, one parallel to $\vec{j}$, and one parallel to $\vec{k}$. 2. **Formula and explanation:** Any vector in 3D space can be expressed as the sum of its components along the unit vectors $\vec{i}$, $\vec{j}$, and $\vec{k}$. The components parallel to these unit vectors are found by projecting $\vec{v}$ onto each unit vector. 3. **Calculate the components:** - Component parallel to $\vec{i}$ is $3\vec{i}$ because the coefficient of $\vec{i}$ in $\vec{v}$ is 3. - Component parallel to $\vec{j}$ is $5\vec{j}$ because the coefficient of $\vec{j}$ in $\vec{v}$ is 5. - Component parallel to $\vec{k}$ is $1\vec{k}$ because the coefficient of $\vec{k}$ in $\vec{v}$ is 1. 4. **Write the vector as sum of components:** $$\vec{v} = 3\vec{i} + 5\vec{j} + 1\vec{k}$$ 5. **Explanation:** Each component vector points along one of the coordinate axes and their sum gives the original vector. This is a fundamental property of vectors in Cartesian coordinates. **Final answer:** The vector $\vec{v}$ is divided into components $3\vec{i}$, $5\vec{j}$, and $\vec{k}$ parallel to $\vec{i}$, $\vec{j}$, and $\vec{k}$ respectively.