To solve this problem, let's analyze the information given step by step:

Problem:

Find the value(s) of $a$ that make $\vec{v} = 5a \hat{i} - 3 \hat{j}$ parallel to $\vec{w} = a^2 \hat{i} + 9 \hat{j}$.

Step 1: Condition for parallel vectors

Two vectors $\vec{v}$ and $\vec{w}$ are parallel if and only if their components satisfy: $\frac{\text{Component along } \hat{i}}{\text{Component along } \hat{j}} = \text{constant}.$ For $\vec{v} = 5a \hat{i} - 3 \hat{j}$ and $\vec{w} = a^2 \hat{i} + 9 \hat{j}$, this implies: $\frac{5a}{-3} = \frac{a^2}{9}.$

Step 2: Solve for $a$

Rewriting the equation: $\frac{5a}{-3} = \frac{a^2}{9}.$ Cross-multiply: $9(5a) = -3(a^2).$ Simplify: $45a = -3a^2.$ Rearrange to form a quadratic equation: $3a^2 + 45a = 0.$ Factorize: $3a(a + 15) = 0.$

Step 3: Solve for roots

From the factored form: $3a = 0 \quad \text{or} \quad a + 15 = 0.$ This gives: $a = 0 \quad \text{or} \quad a = -15.$

Step 4: Verify the solutions

To ensure the solutions make the vectors parallel:

1. When $a = 0$: $\vec{v} = 0\hat{i} - 3\hat{j}, \quad \vec{w} = 0\hat{i} + 9\hat{j}.$ The vectors are not parallel (one is vertical while the other is horizontal). Thus, $a = 0$ is not valid.

2. When $a = -15$: $\vec{v} = -75\hat{i} - 3\hat{j}, \quad \vec{w} = 225\hat{i} + 9\hat{j}.$ Checking $\frac{-75}{-3} = \frac{225}{9}$, which simplifies to $25 = 25$. The vectors are parallel for $a = -15$.

Final Answer:

$\boxed{-15}$

Would you like more details or further explanation?
Here are 5 related questions to expand on this topic:

1. What are the conditions for two vectors to be perpendicular instead of parallel?
2. How would the solution change if the vectors were not 2D but 3D?
3. Can two zero vectors be considered parallel?
4. How can we find the angle between two vectors using their components?
5. What is the physical significance of parallel vectors in real-world applications?

Tip: Always check for invalid solutions by verifying all conditions of the problem after solving equations!