The value of x and y for which vector `A= (6hati+xhatj-2hatk)and B=(5hati+6hatj-yhatk)` may be parallel are

Options

A. `x=0,y=(2)/(3)`
B. `x=(36)/(5),y=(5)/(3)`
C. `x=(15)/(3),y=(23)/(3)`
D. `x=(36)/(5),y=(15)/(4)`

Explanation

To find the values of \( x \) and \( y \) for which the vectors \( \mathbf{A} = 6 \hat{i} + x \hat{j} - 2 \hat{k} \) and \( \mathbf{B} = 5 \hat{i} + 6 \hat{j} - y \hat{k} \) are parallel, we can use the property that two vectors are parallel if the ratios of their corresponding components are equal.

### Step-by-step Solution:

1. **Write down the vectors**:
\[ \mathbf{A} = 6 \hat{i} + x \hat{j} - 2 \hat{k} \]
\[ \mathbf{B} = 5 \hat{i} + 6 \hat{j} - y \hat{k} \]

2. **Set up the ratios for parallel vectors**:
For vectors \( \mathbf{A} \) and \( \mathbf{B} \) to be parallel, the following conditions must hold:
\[ \frac{6}{5} = \frac{x}{6} = \frac{-2}{-y} \]

3. **Solve for \( x \)**:
From the first ratio:
\[ \frac{6}{5} = \frac{x}{6} \]
Cross-multiplying gives:
\[ 6 \cdot 6 = 5x \implies 36 = 5x \implies x = \frac{36}{5} \]

4. **Solve for \( y \)**:
From the second ratio:
\[ \frac{6}{5} = \frac{-2}{-y} \]
This simplifies to:
\[ \frac{6}{5} = \frac{2}{y} \]
Cross-multiplying gives:
\[ 6y = 10 \implies y = \frac{10}{6} = \frac{5}{3} \]

5. **Final values**:
The values of \( x \) and \( y \) for which the vectors are parallel are:
\[ x = \frac{36}{5}, \quad y = \frac{5}{3} \]

### Summary:
Thus, the final answer is:
\[ x = \frac{36}{5}, \quad y = \frac{5}{3} \]