https://file.aiquickdraw.com/mathbot/file/1729791584877k8rw9jr1.jpg

The problem in the image asks to find the distance \(\rho\) between two skew lines \(l_1\) and \(l_2\) using the given formula options.

We are given the parametric equations of the two lines:

\[
l_1: \frac{x - x_1}{m_1} = \frac{y - y_1}{n_1} = \frac{z - z_1}{p_1}
\]
\[
l_2: \frac{x - x_2}{m_2} = \frac{y - y_2}{n_2} = \frac{z - z_2}{p_2}
\]

In 3D space, the distance \(\rho\) between two skew lines can be found using the following determinant-based formula:

\[
\rho = \frac{| (\vec{s_2} - \vec{s_1}) \cdot (\vec{M_1M_2} \times \vec{s_1}) |}{|\vec{s_1} \times \vec{s_2}|}
\]

Where:
- \(\vec{s_1} = (m_1, n_1, p_1)\) is the direction vector of the first line.
- \(\vec{s_2} = (m_2, n_2, p_2)\) is the direction vector of the second line.
- \(\vec{M_1M_2}\) is the vector connecting points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), i.e., \(\vec{M_1M_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\).

Looking at the options, the correct formula is the one involving the cross product of \(\vec{s_1}\) and \(\vec{s_2}\) in the denominator, and the scalar triple product involving \(\vec{s_1}, \vec{s_2},\) and \(\vec{M_1M_2}\) in the numerator. This corresponds to the last option:

\[
\rho = \frac{| \langle \vec{s_1}, \vec{s_2}, \vec{M_1M_2} \rangle |}{| \vec{s_1} \times \vec{s_2} |}
\]

Let me know if you want further explanation.

### Here are 5 questions to deepen understanding:

1. What is the geometric meaning of a scalar triple product?
2. How can you compute the cross product of two vectors in 3D space?
3. Why is the distance between two skew lines given by the magnitude of a scalar triple product?
4. How do you parametrize lines in 3D space from given points and direction vectors?
5. What are the differences between skew lines, parallel lines, and intersecting lines in 3D?

### Tip:
The scalar triple product is useful for finding volumes of parallelepipeds, and in this case, it helps measure the perpendicular distance between two non-intersecting lines.

What is the distance between two skew lines using the given parametric equations?

Learn how to calculate the distance between two skew lines in 3D space using the scalar triple product and cross product. This problem involves parametric equations, direction vectors, and vector operations, suitable for students in Grades 11-12.

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