Finding the orthogonal projection of a vector I can't figure this out. How the book explains to do this problem doesn't make sense to me. I'm hoping someone can simplify this so I can understand how to do this step by step. 
Let $L$ be the line in $\mathbb{R}^3$ that consists of all scalar multiples of $$ \left[
    \begin{array}{ccc}
      2\\
      1\\ 
      2\\
    \end{array}
\right] $$ Find the reflection of the vector 
$$ \left[
    \begin{array}{ccc}
      1\\
      4\\
      1\\
    \end{array}
\right] $$ onto $L$

Let $L$ be the line spanned by some non-zero vector $\mathbf{v} \in \mathbb{R}^3$, so that $L = \{a\mathbf{v} \mid a \in \mathbb{R}\}$ is the space of all scalar multiples of $\mathbf{v}$. Then the orthogonal projection of a vector $\mathbf{x} \in \mathbb{R}^3$ onto the line $L$ can be computed as
$$
 \operatorname{Proj}_L(\mathbf{x}) = \frac{\mathbf{v} \cdot \mathbf{x}}{\mathbf{v}\cdot\mathbf{v}} \mathbf{v}.
$$
So, in this case, we have 
$$
 \mathbf{v} = \begin{pmatrix}2\\1\\2\end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix}1\\4\\1\end{pmatrix},
$$
so that
$$
 \mathbf{v} \cdot \mathbf{x} = 2 \cdot 1 + 1 \cdot 4 + 2 \cdot 1 = 8, \quad \mathbf{v} \cdot \mathbf{v} = 2^2 + 1^2 + 2^2 = 9,
$$
and hence
$$
 \operatorname{Proj}_L(\mathbf{x}) = \frac{8}{9}\begin{pmatrix}2\\1\\2\end{pmatrix}.
$$
Now, you probably wanted to compute the orthogonal projection of some vector $\mathbf{x}$ onto the line $L$ spanned by some non-zero vector $\mathbf{v}$. But what if you did want to compute the reflection of $\mathbf{x}$ in the line $L$? What would this mean? Well, in general, suppose that $S$ is a subspace of $\mathbb{R}^3$ (e.g., a line through the origin or a plane through the origin), so that for any vector $\mathbf{x} \in \mathbb{R}^3$, we have
$$
 \mathbf{x} = \operatorname{Proj}_S(\mathbf{x}) + \operatorname{Proj}_{S^\perp}(\mathbf{x}),
$$
where $\operatorname{Proj}_S(\mathbf{x})$ is the orthogonal projection of $\mathbf{x}$ onto $S$ and $\operatorname{Proj}_{S^\perp}(\mathbf{x})$ is the orthogonal projection of $\mathbf{x}$ onto the orthogonal complement
$$
 S^\perp = \{\mathbf{y} \in \mathbb{R}^3 \mid \text{for every $\mathbf{z} \in S$}, \; \mathbf{y} \cdot \mathbf{z} = 0\}
$$
of $S$; in particular, observe that
$$
 \operatorname{Proj}_{S^\perp}(\mathbf{x}) = \mathbf{x} - \operatorname{Proj}_S(\mathbf{x}).
$$
Then, geometrically, the reflection $\operatorname{Refl}_S(\mathbf{x})$ of $\mathbf{x}$ in $S$ is given by fixing the component of $\mathbf{x}$ in $S$ and flipping the direction of the component of $\mathbf{x}$ in $S^\perp$, i.e.,
$$
 \operatorname{Refl}_S(\mathbf{x}) = \operatorname{Proj}_S(\mathbf{x}) - \operatorname{Proj}_{S^\perp}(\mathbf{x}) = \operatorname{Proj}_S(\mathbf{x}) - \left( \mathbf{x} - \operatorname{Proj}_S(\mathbf{x})\right) = 2\operatorname{Proj}_S(\mathbf{x}) - \mathbf{x}.
$$
So, suppose that $L$ is the line spanned by some non-zero vector $\mathbf{v} \in \mathbb{R}^3$. On the one hand, the reflection of $\mathbf{x}$ in $L$ is given by
$$
 \operatorname{Refl}_L(\mathbf{x}) = 2\operatorname{Proj}_L(\mathbf{x}) - \mathbf{x} = 2\frac{\mathbf{v} \cdot \mathbf{x}}{\mathbf{v}\cdot\mathbf{v}} \mathbf{v} - \mathbf{x},
$$
which in your case yields
$$
 \operatorname{Refl}_L(\mathbf{x}) = 2 \cdot \frac{8}{9}\begin{pmatrix}2\\1\\2\end{pmatrix} - \begin{pmatrix}1\\4\\1\end{pmatrix} = \frac{1}{9}\begin{pmatrix}23\\-20\\23\end{pmatrix}.
$$
On the other hand, the reflection of $\mathbf{x}$ in the plane
$$
 L^\perp = \{\mathbf{y} \in \mathbb{R}^3 \mid \mathbf{v} \cdot \mathbf{y} = 0\}
$$
with normal vector $\mathbf{v}$ is given by
$$
 \operatorname{Refl}_{L^\perp}(\mathbf{x}) = 2\operatorname{Proj}_{L^\perp}(\mathbf{x}) - \mathbf{x} = 2\left(\mathbf{x} - \operatorname{Proj}_L(\mathbf{x})\right) - \mathbf{x} = \mathbf{x} - 2\operatorname{Proj}_L(\mathbf{x}) = \mathbf{x} - 2\frac{\mathbf{v} \cdot \mathbf{x}}{\mathbf{v}\cdot\mathbf{v}} \mathbf{v},
$$
which in your case yields
$$
 \operatorname{Refl}_{L^\perp}(\mathbf{x}) = \begin{pmatrix}1\\4\\1\end{pmatrix} - 2 \cdot \frac{8}{9}\begin{pmatrix}2\\1\\2\end{pmatrix} = -\frac{1}{9}\begin{pmatrix}23\\-20\\23\end{pmatrix}.
$$
One last cultural note: the reflection $\operatorname{Refl}_{L^\perp}(\mathbf{x})$ of $\mathbf{x}$ in the plane $L^\perp$ with normal vector $\mathbf{v}$ is better known in more advanced contexts by another name, namely as the image
$$
 H_{\mathbf{v}}(\mathbf{x}) := \operatorname{Refl}_{L^\perp}(\mathbf{x}) = \mathbf{x} - 2\frac{\mathbf{v} \cdot \mathbf{x}}{\mathbf{v}\cdot\mathbf{v}} \mathbf{v}
$$
of $\mathbf{x}$ under the Householder transformation $H_\mathbf{v}$ corresponding to $\mathbf{v}$. So, in your case, as we just saw,
$$
 H_{\mathbf{v}}(\mathbf{x}) = -\frac{1}{9}\begin{pmatrix}23\\-20\\23\end{pmatrix}.
$$
