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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.

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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 $\alpha$.

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  • $\begingroup$ I did it with wolframaplha and got something complicated, with pi and arccos and I dont know how should i put that in my formula? $\endgroup$ – Ana Matijanovic May 8 '16 at 15:29
  • $\begingroup$ What exactly is the problem? $\endgroup$ – ಠ_ಠ May 18 '16 at 11:47
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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.

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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, $P=I-uu^T$.

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