The vector(s) which is/are coplanar with vectors `hat(i)+hat(j)+2hat(k) and hat(i)+2hat(j)+hat(k)` are perpendicular to the vector `hat(i)+hat(j)+hat(k)` is are

Options

A. `hat(j)-hat(k)`
B. `-hat(i)+hat(j)`
C. `hat(i)-hat(j)`
D. `-hat(j)+hat(k)`

Answer

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 are also perpendicular to the vector \(\hat{i} + \hat{j} + \hat{k}\).

### Step-by-Step Solution:

1. **Define the Vectors:**
Let:
\[ \mathbf{A} = \hat{i} + \hat{j} + 2\hat{k} \]
\[ \mathbf{B} = \hat{i} + 2\hat{j} + \hat{k} \]
We need to find a vector \(\mathbf{R}\) that is coplanar with \(\mathbf{A}\) and \(\mathbf{B}\) and is perpendicular to \(\mathbf{C} = \hat{i} + \hat{j} + \hat{k}\).

2. **Express the Coplanar Vector:**
A vector \(\mathbf{R}\) that is coplanar with \(\mathbf{A}\) and \(\mathbf{B}\) can be expressed as a linear combination of \(\mathbf{A}\) and \(\mathbf{B}\):
\[ \mathbf{R} = \lambda \mathbf{A} + \mu \mathbf{B} \]
Substituting \(\mathbf{A}\) and \(\mathbf{B}\):
\[ \mathbf{R} = \lambda (\hat{i} + \hat{j} + 2\hat{k}) + \mu (\hat{i} + 2\hat{j} + \hat{k}) \]

3. **Combine the Components:**
Combining the components of \(\mathbf{R}\):
\[ \mathbf{R} = (\lambda + \mu)\hat{i} + (\lambda + 2\mu)\hat{j} + (2\lambda + \mu)\hat{k} \]

4. **Condition for Perpendicularity:**
For \(\mathbf{R}\) to be perpendicular to \(\mathbf{C}\), the dot product must equal zero:
\[ \mathbf{R} \cdot \mathbf{C} = 0 \]
Thus, we have:
\[ (\lambda + \mu) + (\lambda + 2\mu) + (2\lambda + \mu) = 0 \]
Simplifying this:
\[ 4\lambda + 4\mu = 0 \]
Therefore:
\[ \lambda + \mu = 0 \quad \Rightarrow \quad \mu = -\lambda \]

5. **Substitute Back:**
Substitute \(\mu = -\lambda\) into the expression for \(\mathbf{R}\):
\[ \mathbf{R} = \lambda (\hat{i} + \hat{j} + 2\hat{k}) - \lambda (\hat{i} + 2\hat{j} + \hat{k}) \]
Simplifying:
\[ \mathbf{R} = \lambda \left( (\hat{i} - \hat{i}) + (\hat{j} - 2\hat{j}) + (2\hat{k} - \hat{k}) \right) \]
\[ \mathbf{R} = \lambda (0\hat{i} - \hat{j} + \hat{k}) = \lambda (-\hat{j} + \hat{k}) \]

6. **Identify the Vectors:**
The vector \(\mathbf{R}\) can be expressed as:
\[ \mathbf{R} = \lambda (-\hat{j} + \hat{k}) \]
For \(\lambda = 1\), we have \(-\hat{j} + \hat{k}\) and for \(\lambda = -1\), we have \(\hat{j} - \hat{k}\).

### 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:
- \(-\hat{j} + \hat{k}\) (Option 4)
- \(\hat{j} - \hat{k}\) (Option 1)