Which one of the following is the unit vector perpendicular to the vectors `4hat(i)+2hat(j) and -3hat(i)+2hat(j)`?

Options

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

Explanation

To find the unit vector that is perpendicular to the vectors \( \mathbf{A} = 4\hat{i} + 2\hat{j} \) and \( \mathbf{B} = -3\hat{i} + 2\hat{j} \), we can follow these steps:

### Step 1: Understand the Condition for Perpendicular Vectors
Two vectors are perpendicular if their dot product is zero. Therefore, we need to find a vector \( \mathbf{R} = x\hat{i} + y\hat{j} + z\hat{k} \) such that:
1. \( \mathbf{R} \cdot \mathbf{A} = 0 \)
2. \( \mathbf{R} \cdot \mathbf{B} = 0 \)

### Step 2: Set Up the Dot Product Equations
Using the vectors given:
1. For \( \mathbf{A} \):
\[ \mathbf{R} \cdot \mathbf{A} = (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (4\hat{i} + 2\hat{j}) = 4x + 2y = 0 \]

2. For \( \mathbf{B} \):
\[ \mathbf{R} \cdot \mathbf{B} = (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (-3\hat{i} + 2\hat{j}) = -3x + 2y = 0 \]

### Step 3: Solve the System of Equations
Now we have the following system of equations:
1. \( 4x + 2y = 0 \) (Equation 1)
2. \( -3x + 2y = 0 \) (Equation 2)

From Equation 1:
\[ 2y = -4x \implies y = -2x \]

Substituting \( y = -2x \) into Equation 2:
\[ -3x + 2(-2x) = 0 \implies -3x - 4x = 0 \implies -7x = 0 \implies x = 0 \]

Substituting \( x = 0 \) back into \( y = -2x \):
\[ y = -2(0) = 0 \]

### Step 4: Determine the Value of z
Since both \( x \) and \( y \) are zero, \( z \) can be any value. Let's take \( z = 1 \) for simplicity. Thus, the vector \( \mathbf{R} \) can be expressed as:
\[ \mathbf{R} = 0\hat{i} + 0\hat{j} + 1\hat{k} = \hat{k} \]

### Step 5: Find the Unit Vector
The vector \( \hat{k} \) is already a unit vector. Therefore, the unit vector perpendicular to the given vectors is:
\[ \hat{k} \]

### Conclusion
The unit vector perpendicular to the vectors \( 4\hat{i} + 2\hat{j} \) and \( -3\hat{i} + 2\hat{j} \) is \( \hat{k} \).