Which one of the following vectors is normal to the vector `hat(i) + hat(j) + hat(k)` ?

Options

A. `hat(i) + hat(j)- hat(k)`
B. `hat(i) - hat(j) + hat(k)`
C. `hat(i) - hat(j) - hat(k)`
D. none of these

Answer

To determine which vector is normal to the vector \( \hat{i} + \hat{j} + \hat{k} \), we need to find a vector \( \mathbf{n} \) such that the dot product of \( \mathbf{a} \) and \( \mathbf{n} \) is zero.

### Step-by-Step Solution:

1. **Identify the given vector**:
The vector \( \mathbf{a} \) is given as:
\[ \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \]

2. **Understand the condition for normal vectors**:
A vector \( \mathbf{n} \) is normal to \( \mathbf{a} \) if:
\[ \mathbf{a} \cdot \mathbf{n} = 0 \]
This means that the dot product of \( \mathbf{a} \) and \( \mathbf{n} \) must equal zero.

3. **Consider the options**:
Let's denote the options as follows:
- Option A: \( \hat{i} + \hat{j} - \hat{k} \)
- Option B: \( \hat{i} - \hat{j} + \hat{k} \)
- Option C: \( \hat{i} - \hat{j} - \hat{k} \)
- Option D: None of these

4. **Calculate the dot product for each option**:
- **Option A**:
\[ \mathbf{n} = \hat{i} + \hat{j} - \hat{k} \]
\[ \mathbf{a} \cdot \mathbf{n} = (1)(1) + (1)(1) + (1)(-1) = 1 + 1 - 1 = 1 \quad (\text{not normal}) \]

- **Option B**:
\[ \mathbf{n} = \hat{i} - \hat{j} + \hat{k} \]
\[ \mathbf{a} \cdot \mathbf{n} = (1)(1) + (1)(-1) + (1)(1) = 1 - 1 + 1 = 1 \quad (\text{not normal}) \]

- **Option C**:
\[ \mathbf{n} = \hat{i} - \hat{j} - \hat{k} \]
\[ \mathbf{a} \cdot \mathbf{n} = (1)(1) + (1)(-1) + (1)(-1) = 1 - 1 - 1 = -1 \quad (\text{not normal}) \]

5. **Conclusion**:
Since none of the options A, B, or C yield a dot product of zero, the correct answer is:
\[ \text{Option D: None of these} \]