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?

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!