Write the value of 'p' for which : `vec(a)=3hat(i)+2hat(j)+9hat(k)` and `vec(b)=hat(i)+p hat(j)+3hat(k)` are parallel.

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 means we can express \(\vec{a}\) as:
\[ \vec{a} = \lambda \vec{b} \]
for some scalar \(\lambda\).

### Step 2: Set up the equation
Given:
\[ \vec{a} = 3\hat{i} + 2\hat{j} + 9\hat{k} \]
\[ \vec{b} = \hat{i} + p\hat{j} + 3\hat{k} \]
We can write:
\[ 3\hat{i} + 2\hat{j} + 9\hat{k} = \lambda (\hat{i} + p\hat{j} + 3\hat{k}) \]

### Step 3: Expand the right side
Expanding the right side gives:
\[ 3\hat{i} + 2\hat{j} + 9\hat{k} = \lambda \hat{i} + \lambda p\hat{j} + 3\lambda \hat{k} \]

### Step 4: 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 5: Solve for \(\lambda\)
From the first equation, we have:
\[ \lambda = 3 \]

### Step 6: Substitute \(\lambda\) into the second equation
Substituting \(\lambda\) into the second equation:
\[ 2 = 3p \]

### Step 7: Solve for \(p\)
Now, solving for \(p\):
\[ p = \frac{2}{3} \]

### Final Answer
Thus, the value of \(p\) for which the vectors are parallel is:
\[ \boxed{\frac{2}{3}} \]