A vector perpendicular to `(4hati-3hatj)` may be :

Options

A. `4hati + 3hatj`
B. `7hatk`
C. `6hati`
D. `3hati-4hatj`

Answer

To find a vector that is perpendicular to the vector \( \mathbf{A} = 4\hat{i} - 3\hat{j} \), we can use the property that the dot product of two perpendicular vectors is zero.

### Step-by-Step Solution:

1. **Understand the Condition for Perpendicularity:**
Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular if their dot product is zero:
\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

2. **Identify the Given Vector:**
The given vector is:
\[ \mathbf{A} = 4\hat{i} - 3\hat{j} \]

3. **Consider a General Vector:**
Let’s denote a general vector \( \mathbf{B} = a\hat{i} + b\hat{j} + c\hat{k} \). We will check the dot product with \( \mathbf{A} \).

4. **Calculate the Dot Product:**
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is:
\[ \mathbf{A} \cdot \mathbf{B} = (4\hat{i} - 3\hat{j}) \cdot (a\hat{i} + b\hat{j} + c\hat{k}) = 4a - 3b + 0c \]
This simplifies to:
\[ 4a - 3b = 0 \]

5. **Finding Possible Values for \( a \) and \( b \):**
Rearranging gives:
\[ 4a = 3b \quad \Rightarrow \quad \frac{a}{b} = \frac{3}{4} \]
This means that \( a \) can be expressed in terms of \( b \) as \( a = \frac{3}{4}b \).

6. **Choosing Specific Values:**
We can choose specific values for \( a \) and \( b \) to find a perpendicular vector. For example, let’s set \( b = 4 \):
\[ a = \frac{3}{4} \times 4 = 3 \]
Thus, one possible vector is:
\[ \mathbf{B} = 3\hat{i} + 4\hat{j} \]

7. **Check for a Vector with a k-component:**
We can also consider a vector with only a k-component, such as \( \mathbf{B} = 7\hat{k} \). The dot product with \( \mathbf{A} \) is:
\[ \mathbf{A} \cdot \mathbf{B} = (4\hat{i} - 3\hat{j}) \cdot (7\hat{k}) = 0 \]
This confirms that \( 7\hat{k} \) is also perpendicular.

8. **Final Answer:**
Therefore, a vector that is perpendicular to \( 4\hat{i} - 3\hat{j} \) is:
\[ \mathbf{B} = 7\hat{k} \]