Two vectors A and B are given as, `A=(3xhati+yhatj)` and `B=(4yhati-3xhatj)`. Find the values of x and y if the the vector `(2A-B)` is `(6hati+9hatj)`.

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the vector \( 2A - B \) equals \( 6\hat{i} + 9\hat{j} \). We start by substituting the given vectors \( A \) and \( B \).

### Step 1: Write down the vectors
Given:
\[ A = 3x\hat{i} + y\hat{j} \]
\[ B = 4y\hat{i} - 3x\hat{j} \]

### Step 2: Compute \( 2A \)
Now, we calculate \( 2A \):
\[ 2A = 2(3x\hat{i} + y\hat{j}) = 6x\hat{i} + 2y\hat{j} \]

### Step 3: Compute \( 2A - B \)
Next, we compute \( 2A - B \):
\[ 2A - B = (6x\hat{i} + 2y\hat{j}) - (4y\hat{i} - 3x\hat{j}) \]
Distributing the negative sign:
\[ 2A - B = 6x\hat{i} + 2y\hat{j} - 4y\hat{i} + 3x\hat{j} \]
Now, combine the \( \hat{i} \) and \( \hat{j} \) components:
\[ 2A - B = (6x - 4y)\hat{i} + (2y + 3x)\hat{j} \]

### Step 4: Set the equation equal to \( 6\hat{i} + 9\hat{j} \)
We know that:
\[ 2A - B = 6\hat{i} + 9\hat{j} \]
This gives us two equations:
1. \( 6x - 4y = 6 \) (Equation 1)
2. \( 2y + 3x = 9 \) (Equation 2)

### Step 5: Solve the equations
Now we will solve these two equations simultaneously.

**From Equation 1:**
\[ 6x - 4y = 6 \]
Rearranging gives:
\[ 6x = 6 + 4y \quad \Rightarrow \quad x = 1 + \frac{2y}{3} \quad \text{(Equation 3)} \]

**Substituting Equation 3 into Equation 2:**
\[ 2y + 3(1 + \frac{2y}{3}) = 9 \]
Expanding this:
\[ 2y + 3 + 2y = 9 \]
Combining like terms:
\[ 4y + 3 = 9 \]
Subtracting 3 from both sides:
\[ 4y = 6 \quad \Rightarrow \quad y = \frac{3}{2} \]

### Step 6: Substitute \( y \) back to find \( x \)
Now substitute \( y = \frac{3}{2} \) back into Equation 3:
\[ x = 1 + \frac{2(\frac{3}{2})}{3} = 1 + 1 = 2 \]

### Final Values
Thus, the values of \( x \) and \( y \) are:
\[ x = 2, \quad y = \frac{3}{2} \]