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Math Help - Reflections

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    Reflections

    Find the matrix of the reflection in the line in that consists of all scalar multiples of the vector 5 6 .
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    Quote Originally Posted by vcf323 View Post
    Find the matrix of the reflection in the line in that consists of all scalar multiples of the vector 5 6 .
    Since your image is messed up lets do it for an arbitrary vector
    \vec{v}=\begin{pmatrix} a \\ b\end{pmatrix}=ae_1+be_2

    So we need to know that transform of the basis vectors e_1,e_2

    First we need to project e_1 on the vector v

    \displaystyle \text{proj}_{v}e_1=\frac{\vec{v} \cdot e_1}{||v||^2}\vec{v}=\frac{a}{a^2+b^2}\vec{v}

    Now to find the the direction vector we subtract

    \frac{a}{a^2+b^2}\vec{v}-e_1

    The above vector takes us from the tip of e_1 onto the line but we want to reflect across it so we need to multiply it by 2. This gives

    \displaystyle T(e_1)=e_1+2\left( \frac{a}{a^2+b^2}\vec{v}-e_1\right)= \frac{2a}{a^2+b^2}(ae_1+be_2)-e_1=\left( \frac{a^2+2b^2}{a^2+b^2}\right)e_1+\left(\frac{2ab  }{a^2+b^2}\right)e_2

    Now just do the exact same thing with e_2
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