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 \( b \) for which 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 use the property that two vectors are parallel if their cross product is zero.

### Step-by-Step Solution:

1. **Write the vectors:**
\[ \mathbf{P} = 2\hat{i} + b\hat{j} + 2\hat{k} \]
\[ \mathbf{Q} = \hat{i} + \hat{j} + \hat{k} \]

2. **Set up the cross product:**
The cross product \( \mathbf{P} \times \mathbf{Q} \) can be calculated using the determinant of a matrix:
\[ \mathbf{P} \times \mathbf{Q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & b & 2 \\ 1 & 1 & 1 \end{vmatrix} \]

3. **Calculate the determinant:**
Expanding this determinant:
\[ \mathbf{P} \times \mathbf{Q} = \hat{i} \begin{vmatrix} b & 2 \\ 1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 2 \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & b \\ 1 & 1 \end{vmatrix} \]

Now calculating each of these 2x2 determinants:
- For \( \hat{i} \):
\[ \begin{vmatrix} b & 2 \\ 1 & 1 \end{vmatrix} = b \cdot 1 - 2 \cdot 1 = b - 2 \]
- For \( \hat{j} \):
\[ \begin{vmatrix} 2 & 2 \\ 1 & 1 \end{vmatrix} = 2 \cdot 1 - 2 \cdot 1 = 0 \]
- For \( \hat{k} \):
\[ \begin{vmatrix} 2 & b \\ 1 & 1 \end{vmatrix} = 2 \cdot 1 - b \cdot 1 = 2 - b \]

Therefore, the cross product becomes:
\[ \mathbf{P} \times \mathbf{Q} = (b - 2)\hat{i} - 0\hat{j} + (2 - b)\hat{k} \]
Simplifying, we have:
\[ \mathbf{P} \times \mathbf{Q} = (b - 2)\hat{i} + (2 - b)\hat{k} \]

4. **Set the cross product to zero:**
For the vectors to be parallel, the cross product must equal the zero vector:
\[ (b - 2)\hat{i} + (2 - b)\hat{k} = 0 \]

This gives us two equations:
- From the \( \hat{i} \) component: \( b - 2 = 0 \)
- From the \( \hat{k} \) component: \( 2 - b = 0 \)

5. **Solve the equations:**
From \( b - 2 = 0 \):
\[ b = 2 \]
From \( 2 - b = 0 \):
\[ b = 2 \]

Both equations give the same solution.

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