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The problem asks for the following:

1. **Orthogonal projection matrix \( P \)**: This matrix projects a vector \( \mathbf{v} \) onto a plane. The plane is defined by \( \{ x : \mathbf{v} \cdot \mathbf{x} = 0 \} \), meaning the normal to the plane is the vector \( \mathbf{v} \).

2. **Reflection matrix \( R \)**: This matrix reflects a vector with respect to the plane mentioned above.

The general formulas for projection and reflection matrices, when reflecting or projecting onto a plane defined by a vector \( \mathbf{n} \), are given as:

- **Projection matrix \( P \)**:
  \[
  P = I - \frac{\mathbf{v} \mathbf{v}^T}{\mathbf{v} \cdot \mathbf{v}}
  \]
  Here, \( \mathbf{v} \mathbf{v}^T \) is the outer product of the vector \( \mathbf{v} \), and \( \mathbf{v} \cdot \mathbf{v} \) is the dot product (or the squared magnitude of \( \mathbf{v} \)).

- **Reflection matrix \( R \)**:
  \[
  R = I - 2 \frac{\mathbf{v} \mathbf{v}^T}{\mathbf{v} \cdot \mathbf{v}}
  \]
  The reflection matrix flips the vector across the plane.

Let’s solve the problem in parts:

### 1. **Projection Matrix \( P \) for \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \):**
We will first compute the projection matrix. Since the vector \( \mathbf{v} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \) is provided in the question, we can apply the formula for \( P \).

### 2. **Reflection Matrix \( R \) for \( \mathbf{v} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \):**
Next, we will compute the reflection matrix using the same vector.

Let me now compute the projection matrix \( P \) and the reflection matrix \( R \), and verify the results for \( \mathbf{v} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \).Here are the results for the orthogonal projection matrix \( P \) and the reflection matrix \( R \) with respect to the plane, as well as their application to the vector \( \mathbf{v} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \):

### 1. **Projection Matrix \( P \)**:
\[
P = \begin{pmatrix}
0.6667 & -0.3333 & -0.3333 \\
-0.3333 & 0.6667 & -0.3333 \\
-0.3333 & -0.3333 & 0.6667
\end{pmatrix}
\]

### 2. **Reflection Matrix \( R \)**:
\[
R = \begin{pmatrix}
0.3333 & -0.6667 & -0.6667 \\
-0.6667 & 0.3333 & -0.6667 \\
-0.6667 & -0.6667 & 0.3333
\end{pmatrix}
\]

### 3. **Verification for \( \mathbf{v} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \)**:

- \( P \mathbf{v} \approx \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \), indicating that the vector is projected onto the plane.
- \( R \mathbf{v} = \begin{pmatrix} -1 \\ -1 \\ -1 \end{pmatrix} \), indicating a reflection of the vector across the plane.

Would you like further details or clarification?

Here are 5 related questions for practice:
1. How would the projection and reflection matrices change if \( \mathbf{v} \) were different?
2. Can you compute the projection of a vector onto a different plane defined by another vector?
3. What is the geometric interpretation of the projection matrix in higher dimensions?
4. How does the reflection matrix affect vectors orthogonal to the plane of reflection?
5. What is the determinant of the reflection matrix, and what does it signify?

**Tip**: The projection matrix always has the property that applying it twice gives the same result (i.e., \( P^2 = P \)). This is called idempotency.

Given a vector v in R^3, compute the matrix P of an orthogonal projection onto the plane {x: v . x = 0} and the matrix R of the reflection with respect to this plane. What is Pv and Rv? Verify the result for v = (1, 1, 1) by multiplication.

Explore the computation of orthogonal projection matrix P and reflection matrix R for vectors in 3-dimensional space, demonstrating through a case study with vector v = (1, 1, 1). This topic is crucial for understanding vector transformations and their geometric interpretations in linear algebra.

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