Find the distance between the parallel lines ` (x)/(2) =(y)/(-1) =(z)/(2) and (x-1)/(2) = (y-1)/(-1) = (z-3)/(2)`.

Answer

To find the distance between the given parallel lines, we will follow these steps:

1. **Identify the equations of the lines**:
The first line is given by:
\[ \frac{x}{2} = \frac{y}{-1} = \frac{z}{2} \]
The second line is given by:
\[ \frac{x-1}{2} = \frac{y-1}{-1} = \frac{z-3}{2} \]

2. **Convert the equations into vector form**:
For the first line, we can express it in vector form as:
\[ \mathbf{r_1} = \mathbf{a_1} + \lambda \mathbf{b} \]
where \(\mathbf{a_1} = (0, 0, 0)\) and \(\mathbf{b} = (2, -1, 2)\).

For the second line, we express it as:
\[ \mathbf{r_2} = \mathbf{a_2} + \mu \mathbf{b} \]
where \(\mathbf{a_2} = (1, 1, 3)\) and \(\mathbf{b} = (2, -1, 2)\).

3. **Calculate the vector \( \mathbf{a_2} - \mathbf{a_1} \)**:
\[ \mathbf{a_2} - \mathbf{a_1} = (1, 1, 3) - (0, 0, 0) = (1, 1, 3) \]

4. **Find the cross product of \( \mathbf{a_2} - \mathbf{a_1} \) and \( \mathbf{b} \)**:
\[ \mathbf{b} = (2, -1, 2) \]
The cross product \( \mathbf{c} = (\mathbf{a_2} - \mathbf{a_1}) \times \mathbf{b} \) can be calculated using the determinant:
\[ \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 3 \\ 2 & -1 & 2 \end{vmatrix} \]

Expanding this determinant:
\[ \mathbf{c} = \mathbf{i} \begin{vmatrix} 1 & 3 \\ -1 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 3 \\ 2 & 2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} \]
\[ = \mathbf{i} (1 \cdot 2 - 3 \cdot (-1)) - \mathbf{j} (1 \cdot 2 - 3 \cdot 2) + \mathbf{k} (1 \cdot (-1) - 1 \cdot 2) \]
\[ = \mathbf{i} (2 + 3) - \mathbf{j} (2 - 6) + \mathbf{k} (-1 - 2) \]
\[ = 5\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} \]

5. **Calculate the magnitude of the cross product**:
\[ |\mathbf{c}| = \sqrt{5^2 + 4^2 + (-3)^2} = \sqrt{25 + 16 + 9} = \sqrt{50} \]

6. **Calculate the magnitude of vector \( \mathbf{b} \)**:
\[ |\mathbf{b}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

7. **Calculate the distance between the two parallel lines**:
The distance \( d \) between the two parallel lines is given by:
\[ d = \frac{|\mathbf{c}|}{|\mathbf{b}|} = \frac{\sqrt{50}}{3} = \frac{5\sqrt{2}}{3} \]

Thus, the distance between the parallel lines is:
\[ \frac{5\sqrt{2}}{3} \text{ units} \]