# Determine reflection matrix over a line

I should determine reflection matrix over a line through the origin with direction vector $\vec{v}=\left(a,b\right) ^{T}$

I dont understand this really good and couldnt find anything helpful on internet. I only found in one book the following:

A is the matrix with reflection over a line through the origin with direction vector $\left(\cos(\frac{\alpha }{2} ) , \sin(\frac{\alpha }{2} ) \right) ^{T}$

$A=\begin{pmatrix} \cos(\alpha ) & \sin(\alpha ) \\ \sin(\alpha ) & -\cos(\alpha ) \end{pmatrix}$

I am not sure how to connect this from book to solve my example, because there is another direction vector. I would be thankful if someone could give me some tips or something like that.

The line is independent of the length of the direction vector, so you can assume without loss of generality that $||(a,b)^t|| = 1$ Because if $||(a,b)^t|| \neq 1$, consider the vector $(a',b')^t := \frac{1}{||(a,b)^t||}(a,b)^t$ that obviously still points in the same direction.
Then set $a=\cos(\alpha/2)$ and $b=\sin(\alpha/2)$ and solve for $\alpha$.
Hint : Draw the cartesian coordinate system ($x'x$ and $y'y$). Then draw a random line and a random vector. Then define geometrically or through analytic geometry the reflection of your given vector and form the matrix of the operator that reflects the vector over a random line.
Assume without loss of generality $v$ is a unit vector and use a unit vector $u=(-b,a)^T$ orthogonal to $v$ to write the reflection matrix which is called Householder matrix, $P=I-uu^T$.