Misshap/typo in question/answer? I am trying to understand question 3. The first image below is some information about how to solve it. Take notice of the of marked yellow area, espicially about how it becomes $1/9$.

Here on question one it states $1/3$ in the answer, that is fine.


This is where I'm confused, it's referring to question one, however it has now changed the formula? The answer of question one gave a certain formula than what is being seen in the question.

And the last part that is weird, why is the answer $1/3$ and not $1/9$ as stated in the first information image?

Is this a typo?
 A: No, it's all consistent. Note that problems $1$ and $3$ involve different vectors compared to the vectors in the first image's answer (in Example $2$, $\textbf a$ and $\textbf b$ swapped), so their answers are unrelated.

Let:
$$
\textbf{a} = \begin{bmatrix}
1 \\ 1 \\ 1
\end{bmatrix}
\qquad\text{and}\qquad
\textbf{b} = \begin{bmatrix}
1 \\ 2 \\ 2
\end{bmatrix}
$$
Then by using the formula from Example $2$, the projection matrix that projects any vector onto $\textbf a$ is given by:
$$
P
= \frac{1}{\textbf a^T \textbf a}\textbf a \textbf a^T
= \frac{1}{1 + 1 + 1}\begin{bmatrix}
1 \\ 1 \\ 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 1
\end{bmatrix}
= \frac{1}{3}\begin{bmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1 \\
\end{bmatrix}
$$
In particular, if we want to project $\textbf b$ onto $\textbf a$, then we have:
\begin{align*}
P\textbf b
&= \left(\frac{1}{\textbf a^T \textbf a}\textbf a \textbf a^T \right)\textbf b \\
&= \frac{1}{\textbf a^T \textbf a}\textbf a (\textbf a^T \textbf b) &\text{by associativity}\\
&= \frac{\textbf a^T \textbf b}{\textbf a^T \textbf a}\textbf a &\text{since $\textbf a^T \textbf b$ is a scalar, so it commutes with $\textbf a$} \\
&= \frac{1 + 2 + 2}{1 + 1 + 1}\textbf a \\
&= \frac{5}{3}\textbf a \\
\end{align*}
