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To find the projection matrix that projects any vector onto the line spanned by \((1, 2, 2)\), let’s go through the solution step by step.

### Step 1: Define the Vector to Project Onto
The direction vector for the line is:
\[
\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}
\]

### Step 2: Compute \(\mathbf{u} \cdot \mathbf{u}\)
The projection matrix formula requires \(\mathbf{u} \cdot \mathbf{u}\), which is the dot product of \(\mathbf{u}\) with itself:
\[
\mathbf{u} \cdot \mathbf{u} = 1^2 + 2^2 + 2^2 = 1 + 4 + 4 = 9
\]

### Step 3: Compute the Projection Matrix
The projection matrix \(P\) that projects any vector \(\mathbf{v}\) onto the direction of \(\mathbf{u}\) is given by:
\[
P = \frac{\mathbf{u} \mathbf{u}^T}{\mathbf{u} \cdot \mathbf{u}}
\]

Since \(\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\), we have:
\[
\mathbf{u} \mathbf{u}^T = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} \begin{pmatrix} 1 & 2 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 4 & 4 \\ 2 & 4 & 4 \end{pmatrix}
\]

Now divide by \(\mathbf{u} \cdot \mathbf{u} = 9\):
\[
P = \frac{1}{9} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 4 & 4 \\ 2 & 4 & 4 \end{pmatrix} = \begin{pmatrix} \frac{1}{9} & \frac{2}{9} & \frac{2}{9} \\ \frac{2}{9} & \frac{4}{9} & \frac{4}{9} \\ \frac{2}{9} & \frac{4}{9} & \frac{4}{9} \end{pmatrix}
\]

### Final Answer
The projection matrix is:
\[
P = \begin{pmatrix} \frac{1}{9} & \frac{2}{9} & \frac{2}{9} \\ \frac{2}{9} & \frac{4}{9} & \frac{4}{9} \\ \frac{2}{9} & \frac{4}{9} & \frac{4}{9} \end{pmatrix}
\]

Would you like further clarification on any step, or do you have additional questions?

### Related Questions
1. How do we interpret a projection matrix in a geometric context?
2. What would be the projection of a vector onto a plane rather than a line?
3. How do we use this projection matrix to project a specific vector, such as \((1, 1, 1)\)?
4. What is the significance of a projection matrix being idempotent (i.e., \(P^2 = P\))?
5. How can we generalize this method to find a projection onto any subspace?

### Tip
When projecting a vector onto a line, you can simplify calculations by normalizing the direction vector first. This makes the projection matrix's construction slightly easier.

Project (1, 1, 1) onto the line spanned by (1, 2, 2), find the projection matrix.

Learn how to find the projection matrix for projecting the vector (1, 1, 1) onto the line spanned by (1, 2, 2). This solution includes step-by-step calculations of the projection matrix, covering core linear algebra concepts such as vector projection, dot products, and matrix formulation. Ideal for undergraduate students studying linear algebra.

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