Consider a vector `vec(F)= 4hat(i)-3hat(j)`. Another vector that is perpendicular to `vec(F)` is

Options

A. `4hat(i)+3hat(j)`
B. `6hat(i)`
C. `7hat(k)`
D. `3hat(i)-4hat(j)`

Explanation

To find a vector that is perpendicular to the given vector \(\vec{F} = 4\hat{i} - 3\hat{j}\), we can use the property of the dot product. Two vectors are perpendicular if their dot product equals zero.

### Step-by-Step Solution:

1. **Identify the Given Vector**:
The vector we have is:
\[ \vec{F} = 4\hat{i} - 3\hat{j} \]

2. **Understand the Condition for Perpendicular Vectors**:
For a vector \(\vec{A} = a\hat{i} + b\hat{j} + c\hat{k}\) to be perpendicular to \(\vec{F}\), the dot product must satisfy:
\[ \vec{F} \cdot \vec{A} = 0 \]

3. **Calculate the Dot Product**:
The dot product of \(\vec{F}\) and \(\vec{A}\) is given by:
\[ \vec{F} \cdot \vec{A} = (4\hat{i} - 3\hat{j}) \cdot (a\hat{i} + b\hat{j} + c\hat{k}) = 4a - 3b + 0c \]
For the vectors to be perpendicular, we set the dot product equal to zero:
\[ 4a - 3b = 0 \]

4. **Solve for One Variable in Terms of the Other**:
Rearranging the equation gives:
\[ 4a = 3b \implies a = \frac{3}{4}b \]
This means that for any value of \(b\), we can find a corresponding \(a\).

5. **Choose a Value for \(b\)**:
Let's choose \(b = 4\). Then:
\[ a = \frac{3}{4} \times 4 = 3 \]
Now we can choose \(c\) as any value. Let's set \(c = 0\) for simplicity.

6. **Construct the Perpendicular Vector**:
Thus, one possible vector that is perpendicular to \(\vec{F}\) is:
\[ \vec{A} = 3\hat{i} + 4\hat{j} + 0\hat{k} = 3\hat{i} + 4\hat{j} \]

7. **Verify the Perpendicularity**:
To verify, we can calculate the dot product:
\[ \vec{F} \cdot \vec{A} = (4)(3) + (-3)(4) = 12 - 12 = 0 \]
Since the dot product is zero, \(\vec{A}\) is indeed perpendicular to \(\vec{F}\).

### Final Answer:
One vector that is perpendicular to \(\vec{F} = 4\hat{i} - 3\hat{j}\) is:
\[ \vec{A} = 3\hat{i} + 4\hat{j} \]