Find the values of x and y for which vectors `vecA=6hati+xhatj-2hatk and vecB=5hati-6hatj-yhatk` may be parellel

Options

A. `x=0`, `y=(2)/(3)`
B. `x=(-36)/(5)`, `y=(5)/(3)`
C. `x=(-15)/(3)`, `y=(23)/(5)`
D. `x=(36)/(3)`, `y=(15)/(14)`

Answer

To determine the values of \( x \) and \( y \) for which the vectors \( \vec{A} = 6\hat{i} + x\hat{j} - 2\hat{k} \) and \( \vec{B} = 5\hat{i} - 6\hat{j} - y\hat{k} \) 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**:
\[ \vec{A} = 6\hat{i} + x\hat{j} - 2\hat{k} \]
\[ \vec{B} = 5\hat{i} - 6\hat{j} - y\hat{k} \]

2. **Set up the ratios for parallelism**:
For the vectors to be parallel, the following ratios must hold:
\[ \frac{6}{5} = \frac{x}{-6} = \frac{-2}{-y} \]

3. **Equate the first two ratios**:
From the first two ratios:
\[ \frac{6}{5} = \frac{x}{-6} \]
Cross-multiplying gives:
\[ 6 \cdot (-6) = 5 \cdot x \]
Simplifying this:
\[ -36 = 5x \]
Thus,
\[ x = \frac{-36}{5} \]

4. **Equate the first and third ratios**:
Now, using the first and third ratios:
\[ \frac{6}{5} = \frac{-2}{-y} \]
This simplifies to:
\[ \frac{6}{5} = \frac{2}{y} \]
Cross-multiplying gives:
\[ 6y = 10 \]
Thus,
\[ y = \frac{10}{6} = \frac{5}{3} \]

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