If the vectors `xhati - 3hatj + 7 hatk` and `hati + y hatj- z hatk` are colliner then the value of `(xy^(2))/(z)` is equla .

Options

A. `(9)/(7)`
B. `(-9)/(7)`
C. `(-7)/(9)`
D. `(7)/(9)`

Explanation

To determine the value of \(\frac{xy^2}{z}\) given that the vectors \(\vec{A} = x \hat{i} - 3 \hat{j} + 7 \hat{k}\) and \(\vec{B} = \hat{i} + y \hat{j} - z \hat{k}\) are collinear, we can follow these steps:

### Step 1: Set up the condition for collinearity
Two vectors \(\vec{A}\) and \(\vec{B}\) are collinear if there exists a scalar \(k\) such that:
\[ \vec{A} = k \vec{B} \]
This gives us the following equations based on the components of the vectors:
\[ \frac{x}{1} = \frac{-3}{y} = \frac{7}{-z} = k \]

### Step 2: Express \(x\), \(y\), and \(z\) in terms of \(k\)
From the first part of the equation:
\[ x = k \]
From the second part:
\[ -3 = ky \implies y = \frac{-3}{k} \]
From the third part:
\[ 7 = -kz \implies z = \frac{-7}{k} \]

### Step 3: Substitute \(x\), \(y\), and \(z\) into \(\frac{xy^2}{z}\)
Now we can substitute these expressions into \(\frac{xy^2}{z}\):
\[ \frac{xy^2}{z} = \frac{k \left(\frac{-3}{k}\right)^2}{\frac{-7}{k}} \]

### Step 4: Simplify the expression
Calculating \(y^2\):
\[ y^2 = \left(\frac{-3}{k}\right)^2 = \frac{9}{k^2} \]
Now substituting this back:
\[ \frac{xy^2}{z} = \frac{k \cdot \frac{9}{k^2}}{\frac{-7}{k}} = \frac{9}{k} \cdot \frac{k}{-7} = \frac{9}{-7} = -\frac{9}{7} \]

### Conclusion
Thus, the value of \(\frac{xy^2}{z}\) is:
\[ -\frac{9}{7} \]