To find the equation of the line of shortest distance between the two given lines, we can follow these steps:
### Step 1: Write the equations of the lines in parametric form
The first line is given by:
\[
\frac{x - 1}{-2} = \frac{y + 3}{2} = \frac{z - 4}{-1} = k
\]
From this, we can express the coordinates in terms of \( k \):
\[
x_1 = 1 - 2k, \quad y_1 = 2k - 3, \quad z_1 = 4 - k
\]
The second line is given by:
\[
\frac{x + 3}{6} = \frac{y - 2}{2} = \frac{z + 5}{3} = \lambda
\]
From this, we can express the coordinates in terms of \( \lambda \):
\[
x_2 = 6\lambda - 3, \quad y_2 = 2\lambda + 2, \quad z_2 = 3\lambda - 5
\]
### Step 2: Find the direction vectors of the lines
The direction vector of the first line is:
\[
\mathbf{a_1} = (-2, 2, -1)
\]
The direction vector of the second line is:
\[
\mathbf{a_2} = (6, 2, 3)
\]
### Step 3: Find the vector connecting points on the two lines
The vector connecting points on the two lines is given by:
\[
\mathbf{d} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
\]
Substituting the expressions from Step 1:
\[
\mathbf{d} = (6\lambda - 3 - (1 - 2k), (2\lambda + 2 - (2k - 3)), (3\lambda - 5 - (4 - k))
\]
This simplifies to:
\[
\mathbf{d} = (6\lambda - 2k - 4, 2\lambda + 2k + 1, 3\lambda + k - 9)
\]
### Step 4: Set up the equations for the shortest distance
The shortest distance between two skew lines can be found using the formula:
\[
\text{Distance} = \frac{|(\mathbf{a_1} \times \mathbf{a_2}) \cdot \mathbf{d}|}{|\mathbf{a_1} \times \mathbf{a_2}|}
\]
### Step 5: Calculate the cross product of the direction vectors
Calculate \(\mathbf{a_1} \times \mathbf{a_2}\):
\[
\mathbf{a_1} \times \mathbf{a_2} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-2 & 2 & -1 \\
6 & 2 & 3
\end{vmatrix}
\]
Calculating this determinant gives:
\[
\mathbf{a_1} \times \mathbf{a_2} = (2 \cdot 3 - (-1) \cdot 2)\mathbf{i} - (-2 \cdot 3 - (-1) \cdot 6)\mathbf{j} + (-2 \cdot 2 - 2 \cdot 6)\mathbf{k}
\]
\[
= (6 + 2)\mathbf{i} - (-6 + 6)\mathbf{j} + (-4 - 12)\mathbf{k}
\]
\[
= 8\mathbf{i} + 0\mathbf{j} - 16\mathbf{k} = (8, 0, -16)
\]
### Step 6: Calculate the magnitude of the cross product
The magnitude is:
\[
|\mathbf{a_1} \times \mathbf{a_2}| = \sqrt{8^2 + 0^2 + (-16)^2} = \sqrt{64 + 256} = \sqrt{320} = 8\sqrt{5}
\]
### Step 7: Substitute back to find the shortest distance
Now we substitute the values into the distance formula and simplify to find the shortest distance.
### Step 8: Write the equation of the line of shortest distance
The line of shortest distance can be expressed in parametric form using the points on the two lines and the direction vector of the shortest distance.
### Final Equation
The final equation of the line of shortest distance can be written as:
\[
\mathbf{r} = \mathbf{r_1} + t \mathbf{n}
\]
Where \(\mathbf{r_1}\) is a point on one of the lines and \(\mathbf{n}\) is the direction vector of the shortest distance.
