Reflection vector doesn't match book solution

Question

Let L be the line in R3 that consists of all scalar multiples of [2 1 2]. Find the reflection of the vector [1 1 1] in line L.

I'm calculating this using 2(u dot v) u - v. My solution:

2 ([2 1 2] dot [1 1 1])([2 1 2]) - [1 1 1]
= 2 (5) ([2 1 2]) - [1 1 1]
= [20 10 20] - [1 1 1]
= [19 9 19] 

The correct solution is apparently 1/9 [11 1 11]. What am I missing here?

Answer 1

It's because $\|u\|\neq1$. Therefore, the formula that you should have used is$$2\frac{\langle u,v\rangle}{\|u\|^2}u-v.$$Try it and you will see that you will get the correct answer.

Answer 2

For further clarity, what I was missing here (and what the book failed to mention until three chapters later) was that a unit vector $u$ has a length of 1, that is, $u \cdot u = 1$.

$\begin{bmatrix}2\\1\\2\end{bmatrix}$ needed to be normalised to a unit vector.

After normalising, the unit vector is $\begin{bmatrix}\frac{4}{\sqrt{9}}\\\frac{1}{{\sqrt{9}}}\\\frac{4}{{\sqrt{9}}}\end{bmatrix}$, and the answer resolves.