The vectors which is/are coplanar with vectors `hati+hatj+2hatk and hati+2hatj+hatk` and perpendicular to the vector `hati+hatj+hatk ` is /are (A) `hatj-hatk` (B) `-hati+hatj` (C) `hati-hatj` (D) `-hatj+hatk`

Options

A. `hatj - hatk`
B. `-hati + hatj`
C. `hati -hatj`
D. `-hatj +hatk`

Explanation

To solve the problem, we need to find the vectors that are coplanar with the vectors \( \hat{i} + \hat{j} + 2\hat{k} \) and \( \hat{i} + 2\hat{j} + \hat{k} \), and also perpendicular to the vector \( \hat{i} + \hat{j} + \hat{k} \).

### Step-by-Step Solution:

1. **Define the given vectors**:
Let:
\[ \mathbf{A} = \hat{i} + \hat{j} + 2\hat{k} \]
\[ \mathbf{B} = \hat{i} + 2\hat{j} + \hat{k} \]
\[ \mathbf{C} = \hat{i} + \hat{j} + \hat{k} \]

2. **Find a general vector coplanar with \(\mathbf{A}\) and \(\mathbf{B}\)**:
Any vector \(\mathbf{R}\) that is coplanar with \(\mathbf{A}\) and \(\mathbf{B}\) can be expressed as:
\[ \mathbf{R} = \lambda \mathbf{A} + \mu \mathbf{B} \]
where \(\lambda\) and \(\mu\) are scalars.

3. **Substitute \(\mathbf{A}\) and \(\mathbf{B}\)**:
\[ \mathbf{R} = \lambda (\hat{i} + \hat{j} + 2\hat{k}) + \mu (\hat{i} + 2\hat{j} + \hat{k}) \]
Expanding this gives:
\[ \mathbf{R} = (\lambda + \mu)\hat{i} + (\lambda + 2\mu)\hat{j} + (2\lambda + \mu)\hat{k} \]

4. **Condition for perpendicularity**:
The vector \(\mathbf{R}\) must also be perpendicular to \(\mathbf{C}\). For two vectors to be perpendicular, their dot product must equal zero:
\[ \mathbf{R} \cdot \mathbf{C} = 0 \]
Therefore:
\[ (\lambda + \mu) + (\lambda + 2\mu) + (2\lambda + \mu) = 0 \]
Simplifying this:
\[ 4\lambda + 4\mu = 0 \]
or
\[ \lambda + \mu = 0 \quad \Rightarrow \quad \mu = -\lambda \]

5. **Substituting \(\mu\) back into \(\mathbf{R}\)**:
Substitute \(\mu = -\lambda\) into \(\mathbf{R}\):
\[ \mathbf{R} = \lambda (\hat{i} + \hat{j} + 2\hat{k}) - \lambda (\hat{i} + 2\hat{j} + \hat{k}) \]
This simplifies to:
\[ \mathbf{R} = \lambda \left( \hat{j} - \hat{k} \right) \]

6. **Identifying the vectors**:
The vector \(\mathbf{R}\) can be expressed as:
\[ \mathbf{R} = \lambda (\hat{j} - \hat{k}) \]
This means that \(\hat{j} - \hat{k}\) is one of the vectors that satisfy the conditions.

7. **Check the options**:
- (A) \( \hat{j} - \hat{k} \) is valid.
- (B) \( -\hat{i} + \hat{j} \) does not match.
- (C) \( \hat{i} - \hat{j} \) does not match.
- (D) \( -\hat{j} + \hat{k} \) can be obtained by taking \(\lambda = -1\).

### Final Answer:
The vectors that are coplanar with \( \hat{i} + \hat{j} + 2\hat{k} \) and \( \hat{i} + 2\hat{j} + \hat{k} \), and perpendicular to \( \hat{i} + \hat{j} + \hat{k} \) are:
- (A) \( \hat{j} - \hat{k} \)
- (D) \( -\hat{j} + \hat{k} \)