To find the shortest distance between the two skew lines given by the equations:
1. \(\frac{x-3}{-1} = \frac{y-4}{2} = \frac{z+2}{1}\)
2. \(\frac{x-1}{1} = \frac{y+7}{3} = \frac{z+2}{2}\)
we can follow these steps:
### Step 1: Identify Points and Direction Ratios
From the equations of the lines, we can identify the points and direction ratios.
**For Line 1:**
- Point \(A(3, 4, -2)\)
- Direction ratios \(B(-1, 2, 1)\)
**For Line 2:**
- Point \(C(1, -7, -2)\)
- Direction ratios \(D(1, 3, 2)\)
### Step 2: Calculate the Vector \(A - C\)
The vector \(A - C\) is calculated as follows:
\[
A - C = (3 - 1, 4 - (-7), -2 - (-2)) = (2, 11, 0)
\]
### Step 3: Calculate the Cross Product \(B \times D\)
The cross product \(B \times D\) can be calculated using the determinant of the following matrix:
\[
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
-1 & 2 & 1 \\
1 & 3 & 2
\end{vmatrix}
\]
Calculating this determinant:
\[
B \times D = \hat{i}(2 \cdot 2 - 1 \cdot 3) - \hat{j}(-1 \cdot 2 - 1 \cdot 1) + \hat{k}(-1 \cdot 3 - 2 \cdot 1)
\]
\[
= \hat{i}(4 - 3) - \hat{j}(-2 - 1) + \hat{k}(-3 - 2)
\]
\[
= \hat{i}(1) + \hat{j}(3) - \hat{k}(5)
\]
Thus,
\[
B \times D = (1, 3, -5)
\]
### Step 4: Calculate the Magnitude of the Cross Product
The magnitude of \(B \times D\) is:
\[
|B \times D| = \sqrt{1^2 + 3^2 + (-5)^2} = \sqrt{1 + 9 + 25} = \sqrt{35}
\]
### Step 5: Calculate the Shortest Distance
The formula for the shortest distance \(d\) between two skew lines is given by:
\[
d = \frac{|(A - C) \cdot (B \times D)|}{|B \times D|}
\]
Calculating the dot product \((A - C) \cdot (B \times D)\):
\[
(A - C) \cdot (B \times D) = (2, 11, 0) \cdot (1, 3, -5) = 2 \cdot 1 + 11 \cdot 3 + 0 \cdot (-5) = 2 + 33 + 0 = 35
\]
Now substituting into the distance formula:
\[
d = \frac{|35|}{\sqrt{35}} = \frac{35}{\sqrt{35}} = \sqrt{35}
\]
### Final Answer
The shortest distance between the two skew lines is \(\sqrt{35}\).
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