1. **State the problem:** We are given two vectors $\vec{A} = 5\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ and $\vec{B} = 15\mathbf{i} + a\mathbf{j} - 9\mathbf{k}$. We need to find the value of $a$ so that a given vector relation or condition is met. Since the problem does not explicitly state the relation, let's assume it might be checking if vector $\vec{B}$ is a scalar multiple of $\vec{A}$ (common in such problems). 2. **Check for scalar multiple:** If $\vec{B} = k \vec{A}$ for some scalar $k$, then each component of $\vec{B}$ must be $k$ times the corresponding component of $\vec{A}$. From the $\mathbf{i}$ component: $$15 = 5k \implies k = 3$$ Using $k=3$ for the $\mathbf{j}$ component: $$a = 2k = 2 \times 3 = 6$$ Check $\mathbf{k}$ component: $$-9 = -3k = -3 \times 3 = -9$$ This matches correctly. 3. **Final answer:** The value of $a$ is **6** to make $\vec{B}$ a scalar multiple of $\vec{A}$.