1. **State the problem:** We need to find the scalar value $a$ such that the pair of vectors $\vec{U} = (5, 3, a, 2)$ and $\vec{V} = (1, -3, ...)$ satisfy a certain condition. Since the second vector $\vec{V}$ appears incomplete, we assume the problem asks for a condition like orthogonality (dot product = 0). 2. **Assumption:** If the goal is to find $a$ such that $\vec{U}$ and $\vec{V}$ are orthogonal, we use the dot product formula: $$\vec{U} \cdot \vec{V} = 0$$ 3. **Completing vector $\vec{V}$:** The question gives $\vec{V} = (1, -3$ and is incomplete. Typically, a 4-dimensional vector matches $\vec{U}$. Since options for $a$ are numeric, assume $\vec{V} = (1, -3, b, c)$ with unknowns $b, c$? 4. **Since no values given for the remaining components, we cannot determine $a$ directly.** 5. **Likely the question is truncated or missing data.** Without full vector $\vec{V}$, this cannot be solved. If $\vec{V}$ was $(1,-3,a,...)$, possibly the problem intended to solve for $a$ membership in $\vec{U}$ using conditions like equal dot products or zero dot product. **Please provide the complete vector $\vec{V}$ or full problem statement to proceed.**