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 $P$: