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To solve this problem, let’s go through the given information and find the vector equation of line \( L_2 \) as required.

### Given Data
1. **Equation of \( L_1 \):**
   \[
   \mathbf{r} = \begin{pmatrix} -5 \\ -3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix}
   \]
   where \( \lambda \) is a parameter. Here, the direction vector of \( L_1 \) is:
   \[
   \mathbf{d_1} = \begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix}.
   \]

2. **Conditions for \( L_2 \):**
   - \( L_2 \) passes through the origin (0, 0, 0).
   - \( L_2 \) is perpendicular to \( L_1 \).
   - \( L_2 \) intersects \( L_1 \).

### Solution
To find the vector equation of \( L_2 \), we need a direction vector \( \mathbf{d_2} \) for \( L_2 \) that is perpendicular to \( \mathbf{d_1} \).

Since \( L_2 \) is perpendicular to \( L_1 \), the direction vector \( \mathbf{d_2} \) of \( L_2 \) should satisfy:
\[
\mathbf{d_1} \cdot \mathbf{d_2} = 0
\]
This gives us the perpendicularity condition.

1. **Direction Vector for \( L_2 \):**  
   Let \( \mathbf{d_2} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).
   Then:
   \[
   \begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = 0
   \]
   Expanding this dot product, we get:
   \[
   -x + 2y + 2z = 0
   \]
   This equation represents a constraint on the values of \( x \), \( y \), and \( z \) for \( \mathbf{d_2} \).

2. **Choosing Values for \( \mathbf{d_2} \):**  
   To satisfy this equation, we can select values for \( x \), \( y \), and \( z \) that solve \( -x + 2y + 2z = 0 \). One possible choice is:
   \[
   \mathbf{d_2} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}
   \]
   since:
   \[
   -2 + 2 \cdot 1 + 2 \cdot (-1) = 0
   \]

3. **Equation of \( L_2 \):**
   Since \( L_2 \) passes through the origin, its position vector is:
   \[
   \mathbf{r} = \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}
   \]
   where \( \mu \) is a parameter.

Thus, the vector equation of \( L_2 \) is:
\[
\mathbf{r} = \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}
\]

### Final Answer
The vector equation of \( L_2 \) is:
\[
\mathbf{r} = \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}, \quad \mu \in \mathbb{R}
\]

Would you like more details or have any questions about this solution?

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Here are some related questions you might find interesting:
1. How do we confirm if two lines in three-dimensional space intersect?
2. How can we determine the angle between two lines in three-dimensional space?
3. What is the general method for finding a line perpendicular to a given line in 3D space?
4. How do we find the shortest distance between two skew lines?
5. How does the dot product help in determining perpendicularity in vector equations?

**Tip:** When working with perpendicular vectors, using the dot product is a powerful tool to ensure orthogonality!

A line L1 has equation r = (-5, -3, 2) + λ(-1, 2, 2). A line L2 passing through the origin intersects L1 and is perpendicular to L1. (a) Find a vector equation of L2.

Learn how to find the vector equation of a line passing through the origin that is perpendicular to another line in 3D space. This problem involves vector equations, dot product, and the perpendicularity condition.

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