Consider a vector `vecF=4hati-3hatj`. Another vector that is perpendicular to `vecF` is

Options

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

Answer

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

### Step 1: Write the given vector
The given vector is:
\[ \vec{F} = 4\hat{i} - 3\hat{j} \]

### Step 2: Define a general vector
Let’s define a general vector \(\vec{G} = a\hat{i} + b\hat{j}\), where \(a\) and \(b\) are the components of the vector we want to find.

### Step 3: Set up the dot product equation
For \(\vec{F}\) and \(\vec{G}\) to be perpendicular, their dot product must equal zero:
\[ \vec{F} \cdot \vec{G} = 0 \]

Calculating the dot product:
\[ (4\hat{i} - 3\hat{j}) \cdot (a\hat{i} + b\hat{j}) = 4a - 3b \]

### Step 4: Set the dot product to zero
Setting the dot product equal to zero gives us the equation:
\[ 4a - 3b = 0 \]

### Step 5: Solve for one variable in terms of the other
From the equation \(4a - 3b = 0\), we can express \(b\) in terms of \(a\):
\[ 4a = 3b \implies b = \frac{4}{3}a \]

### Step 6: Choose a value for \(a\)
We can choose any non-zero value for \(a\) to find a specific vector. Let’s choose \(a = 3\):
\[ b = \frac{4}{3} \cdot 3 = 4 \]

### Step 7: Write the perpendicular vector
Substituting the values of \(a\) and \(b\) into \(\vec{G}\):
\[ \vec{G} = 3\hat{i} + 4\hat{j} \]

### Conclusion
Thus, one vector that is perpendicular to \(\vec{F} = 4\hat{i} - 3\hat{j}\) is:
\[ \vec{G} = 3\hat{i} + 4\hat{j} \]