Two vectors `P=2hat I +bhat j +2hat k` and `Q=hat i+hat j+hat k` will be parallel if :

Options

A. b=0
B. b=1
C. b=2
D. b=-4

Explanation

To determine the value of \( b \) such that the vectors \( \mathbf{P} = 2\hat{i} + b\hat{j} + 2\hat{k} \) and \( \mathbf{Q} = \hat{i} + \hat{j} + \hat{k} \) are parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Parallel Vectors**:
Two vectors \( \mathbf{P} \) and \( \mathbf{Q} \) are parallel if there exists a scalar \( n \) such that:
\[ \mathbf{P} = n \mathbf{Q} \]

2. **Expressing the Vectors**:
We can express the vectors as:
\[ \mathbf{P} = 2\hat{i} + b\hat{j} + 2\hat{k} \]
\[ \mathbf{Q} = \hat{i} + \hat{j} + \hat{k} \]

3. **Setting Up the Equation**:
From the parallel condition, we have:
\[ 2\hat{i} + b\hat{j} + 2\hat{k} = n(\hat{i} + \hat{j} + \hat{k}) \]

4. **Expanding the Right Side**:
Expanding the right side gives:
\[ n\hat{i} + n\hat{j} + n\hat{k} \]

5. **Equating Components**:
Now, we equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) from both sides:
- For \( \hat{i} \):
\[ 2 = n \quad \text{(1)} \]
- For \( \hat{j} \):
\[ b = n \quad \text{(2)} \]
- For \( \hat{k} \):
\[ 2 = n \quad \text{(3)} \]

6. **Solving for \( n \)**:
From equations (1) and (3), we find:
\[ n = 2 \]

7. **Finding \( b \)**:
Substitute \( n = 2 \) into equation (2):
\[ b = n = 2 \]

### Conclusion:
Thus, the value of \( b \) such that the vectors \( \mathbf{P} \) and \( \mathbf{Q} \) are parallel is:
\[ \boxed{2} \]