Reflection of a Light Ray I found this problem to be very hard while studying for the exam: 

Let
  $$L: \vec r(t)=<1,-2,3>+t<-5,4,1>, \qquad t \in \mathbb{R}$$
  be a line. Light is traveling along the line $L$ in the direction of increasing $t$ value. Let $\Omega: x=0$ (the $yz$ plane) be a mirror. Find the vector equation of the line $\hat L$ that is the reflection of line $L$ by $\Omega$ 
  Any help would be helpful!

Any help would be helpful!

My solution 
  $$L: \vec r(t)= 1/5<0,-6,16>+t<5,4,1>, \qquad t \in \mathbb{R}$$


In general, to find the equation of the reflection ray, you need to the find 
$1.$ the reflection vector and
$2.$ the intersection point of incident ray with the mirror.
Here, I discuss the general case. However, you may find simpler solution for special cases like the one in your example. 
Suppose the equation of the mirror which is a plane is given by
$$(X-P_1) \cdot n =0 \tag{1}$$
where $X$ is an arbitrary point on your plane and $P_1$ is a given special point and $n$ is also the unit normal of the plane. Next, consider that the equation of the incident ray is given by
$$X=P_2+tV_i \tag{2}$$
where $X$ is an arbitray point on the ray and $P_2$ is a given point on the ray. $V_i$ is a vector in the direction of the incident ray. To find the intersection point, we can simply combine $(1)$ and $(2)$ to get
$$t_0=\frac{(P_1-P_2) \cdot n}{V_i \cdot n} \tag{3}$$
and hence the intersection point $P_0$ will be
$$P_0=P_2+\frac{(P_1-P_2) \cdot n}{V_i \cdot n}V_i \tag{4}$$ 
also, we can decompose $V_i$ in the form 
$$\begin{align}
V_i &= V_{i,n}+V_{i,t} \\
&=(V_i \cdot n)n-(V_i \times n) \times n
\end{align} \tag{5}$$
which are the components normal and tangent to the mirror. When we have a reflection the tangential component of the ray remains the same while the normal component changes sign so the reflection vector will be
$$\begin{align}
V_r &= V_{r,n}+V_{r,t} \\
&=-V_{i,n}+V_{i,t} \\
&=-(V_i \cdot n)n-(V_i \times n) \times n \\
&=-2(V_i \cdot n) n + V_i
\end{align} \tag{6}$$
and hence the equation of the reflection line is
$$X=P_0+tV_r \tag{7}$$
or equivalently
$$\boxed{X=P_2+\frac{(P_1-P_2) \cdot n}{V_i \cdot n}V_i+t[V_i-2(V_i \cdot n)n]}\tag{8}$$
