Determine whether or not the following pairs of lines intersect :
`(x -1)/(2) = (y + 1)/(3) = z , (x + 1)/(5) = (y -2)/(1) =(z -2)/(0)`.

Answer

To determine whether the given pairs of lines intersect, we can follow these steps:

### Step 1: Write the equations of the lines in parametric form
The given lines are:
1. \(\frac{x - 1}{2} = \frac{y + 1}{3} = z\)
2. \(\frac{x + 1}{5} = \frac{y - 2}{1} = \frac{z - 2}{0}\)

From the first line, we can express it in parametric form:
- Let \(t\) be the parameter, then:
- \(x_1 = 1 + 2t\)
- \(y_1 = -1 + 3t\)
- \(z_1 = t\)

From the second line, we can express it in parametric form:
- Let \(s\) be the parameter, then:
- \(x_2 = -1 + 5s\)
- \(y_2 = 2 + s\)
- \(z_2 = 2\)

### Step 2: Identify the points and direction ratios
From the parametric equations:
- For the first line, the point \(A\) is \((1, -1, 0)\) and the direction ratios are \((2, 3, 1)\).
- For the second line, the point \(B\) is \((-1, 2, 2)\) and the direction ratios are \((5, 1, 0)\).

### Step 3: Find the vectors
Let:
- \( \vec{A} = (1, -1, 0) \)
- \( \vec{B} = (-1, 2, 2) \)
- Direction vector of the first line: \( \vec{b} = (2, 3, 1) \)
- Direction vector of the second line: \( \vec{d} = (5, 1, 0) \)

### Step 4: Calculate the vector \( \vec{C} = \vec{B} - \vec{A} \)
\[ \vec{C} = \vec{B} - \vec{A} = (-1 - 1, 2 + 1, 2 - 0) = (-2, 3, 2) \]

### Step 5: Calculate \( \vec{b} \times \vec{d} \)
To find the cross product \( \vec{b} \times \vec{d} \):
\[ \vec{b} \times \vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 1 \\ 5 & 1 & 0 \end{vmatrix} \]
Calculating the determinant:
\[ = \hat{i}(3 \cdot 0 - 1 \cdot 1) - \hat{j}(2 \cdot 0 - 1 \cdot 5) + \hat{k}(2 \cdot 1 - 3 \cdot 5) \]
\[ = \hat{i}(0 - 1) - \hat{j}(0 - 5) + \hat{k}(2 - 15) \]
\[ = -\hat{i} + 5\hat{j} - 13\hat{k} \]
Thus, \( \vec{b} \times \vec{d} = (-1, 5, -13) \).

### Step 6: Calculate the magnitude of \( \vec{b} \times \vec{d} \)
\[ |\vec{b} \times \vec{d}| = \sqrt{(-1)^2 + 5^2 + (-13)^2} = \sqrt{1 + 25 + 169} = \sqrt{195} \]

### Step 7: Calculate the dot product \( \vec{C} \cdot (\vec{b} \times \vec{d}) \)
\[ \vec{C} \cdot (\vec{b} \times \vec{d}) = (-2, 3, 2) \cdot (-1, 5, -13) \]
\[ = (-2)(-1) + (3)(5) + (2)(-13) = 2 + 15 - 26 = -9 \]

### Step 8: Calculate the minimum distance \(d_{min}\)
\[ d_{min} = \frac{|\vec{C} \cdot (\vec{b} \times \vec{d})|}{|\vec{b} \times \vec{d}|} = \frac{|-9|}{\sqrt{195}} = \frac{9}{\sqrt{195}} \]

### Step 9: Conclusion
Since \(d_{min} \neq 0\), the lines do not intersect.