Write the value of p for which `veca = 3 hati + 2hatj + 9hatk and vecb = hati + p hatj + 3hatk` are parallel vectors.

Answer

To find the value of \( p \) for which the vectors \( \vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \) and \( \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k} \) are parallel, we can follow these steps:

### Step 1: Understand the condition for parallel vectors
Two vectors \( \vec{a} \) and \( \vec{b} \) are parallel if one is a scalar multiple of the other. This can be expressed mathematically as:
\[ \vec{a} = \lambda \vec{b} \]
for some scalar \( \lambda \).

### Step 2: Write down the vectors
Given:
\[ \vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \]
\[ \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k} \]

### Step 3: Set up the equation
From the condition of parallelism, we have:
\[ 3 \hat{i} + 2 \hat{j} + 9 \hat{k} = \lambda (\hat{i} + p \hat{j} + 3 \hat{k}) \]

### Step 4: Expand the right-hand side
Distributing \( \lambda \) gives:
\[ 3 \hat{i} + 2 \hat{j} + 9 \hat{k} = \lambda \hat{i} + \lambda p \hat{j} + 3\lambda \hat{k} \]

### Step 5: Equate the coefficients
Now, we equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) from both sides:
1. For \( \hat{i} \): \( 3 = \lambda \)
2. For \( \hat{j} \): \( 2 = \lambda p \)
3. For \( \hat{k} \): \( 9 = 3\lambda \)

### Step 6: Solve for \( \lambda \)
From the first equation, we find:
\[ \lambda = 3 \]

### Step 7: Substitute \( \lambda \) into the second equation
Now substitute \( \lambda = 3 \) into the second equation:
\[ 2 = 3p \]

### Step 8: Solve for \( p \)
To find \( p \), we rearrange the equation:
\[ p = \frac{2}{3} \]

### Conclusion
Thus, the value of \( p \) for which the vectors \( \vec{a} \) and \( \vec{b} \) are parallel is:
\[ \boxed{\frac{2}{3}} \]