Consider a force vector `F=hati+ hatj + hatk ` Another vector perpendicular to F is

Options

A. `4hati +3hatj`
B. `6hati`
C. `2hati -hatj -hatk`
D. `3hati - 4hatj`

Explanation

To find a vector that is perpendicular to the given force vector \( \mathbf{F} = \hat{i} + \hat{j} + \hat{k} \), we can use the property that the dot product of two perpendicular vectors is zero.

### Step-by-Step Solution:

1. **Define the Force Vector**:
\[ \mathbf{F} = \hat{i} + \hat{j} + \hat{k} \]

2. **Choose a General Vector**:
Let’s denote another vector \( \mathbf{A} \) as:
\[ \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]
where \( a_1, a_2, a_3 \) are components of vector \( \mathbf{A} \).

3. **Set Up the Dot Product**:
For \( \mathbf{A} \) to be perpendicular to \( \mathbf{F} \), the dot product must equal zero:
\[ \mathbf{F} \cdot \mathbf{A} = 0 \]

4. **Calculate the Dot Product**:
\[ \mathbf{F} \cdot \mathbf{A} = (\hat{i} + \hat{j} + \hat{k}) \cdot (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) = a_1 + a_2 + a_3 \]

5. **Set the Equation to Zero**:
\[ a_1 + a_2 + a_3 = 0 \]

6. **Choose Specific Values for \( a_1, a_2, a_3 \)**:
We can choose specific values that satisfy this equation. For example:
- Let \( a_1 = 1 \)
- Let \( a_2 = -1 \)
- Let \( a_3 = 0 \)

Thus, we have:
\[ \mathbf{A} = 1 \hat{i} - 1 \hat{j} + 0 \hat{k} = \hat{i} - \hat{j} \]

7. **Verify the Perpendicularity**:
Now, we check if \( \mathbf{F} \cdot \mathbf{A} = 0 \):
\[ \mathbf{F} \cdot \mathbf{A} = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j}) = 1 - 1 + 0 = 0 \]
This confirms that \( \mathbf{A} \) is indeed perpendicular to \( \mathbf{F} \).

### Conclusion:
The vector \( \mathbf{A} = \hat{i} - \hat{j} \) is perpendicular to the force vector \( \mathbf{F} = \hat{i} + \hat{j} + \hat{k} \).