A vector perpendicular to `hat(i)+hat(j)+hat(k)` is

Options

A. `hat(i)-hat(j)+hat(k)`
B. `hat(i)-hat(j)-hat(k)`
C. `-hat(i)-hat(j)-hat(k)`
D. `3hat(i)+2hat(j)-5hat(k)`

Explanation

To find a vector that is perpendicular to the vector \( \hat{i} + \hat{j} + \hat{k} \), we can use the property of the dot product. Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular if their dot product is zero, i.e., \( \mathbf{A} \cdot \mathbf{B} = 0 \).

### Step-by-Step Solution:

1. **Identify the Given Vector:**
The vector we are given is:
\[ \mathbf{A} = \hat{i} + \hat{j} + \hat{k} \]

2. **Choose Options for Vector \( \mathbf{B} \):**
We will check each of the given options to see if they are perpendicular to \( \mathbf{A} \).

3. **Check Option 1: \( \mathbf{B} = \hat{i} - \hat{j} + \hat{k} \)**
- Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j} + \hat{k}) = 1 - 1 + 1 = 1 \]
- Since \( \mathbf{A} \cdot \mathbf{B} \neq 0 \), this option is incorrect.

4. **Check Option 2: \( \mathbf{B} = \hat{i} - \hat{j} - \hat{k} \)**
- Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j} - \hat{k}) = 1 - 1 - 1 = -1 \]
- Since \( \mathbf{A} \cdot \mathbf{B} \neq 0 \), this option is also incorrect.

5. **Check Option 3: \( \mathbf{B} = -\hat{i} - \hat{j} - \hat{k} \)**
- Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot (-\hat{i} - \hat{j} - \hat{k}) = -1 - 1 - 1 = -3 \]
- Since \( \mathbf{A} \cdot \mathbf{B} \neq 0 \), this option is incorrect.

6. **Check Option 4: \( \mathbf{B} = 3\hat{i} + 2\hat{j} - 5\hat{k} \)**
- Calculate the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot (3\hat{i} + 2\hat{j} - 5\hat{k}) = 3 + 2 - 5 = 0 \]
- Since \( \mathbf{A} \cdot \mathbf{B} = 0 \), this option is correct.

### Conclusion:
The vector that is perpendicular to \( \hat{i} + \hat{j} + \hat{k} \) is:
\[ \mathbf{B} = 3\hat{i} + 2\hat{j} - 5\hat{k} \]