Q:

Find the normal vectors of the two angle-bisecting planes.

Compute the coordinate equation of the angle bisectors of the planes E
  and F.
$E: x + 4y + 8z + 50 = 0 $   and $F: 3x + 4y + 12z + 82 = 0$
Proceed as follows:
a) Find the normal vectors of the two angle-bisecting planes.
b) Find a shared point of planes E and F.
c) Now determine the equations of the two angle-bisecting planes.

I have the solutions but I don't understand why I must do things the way the solution is shown.
a) $\left|\begin{pmatrix}1\\ 4\\ 8\end{pmatrix}\right|=9$  and  $\left|\begin{pmatrix}3\\ 4\\ 12\end{pmatrix}\right|=13$
are the normal vectors from the equations. But this is not a good enough answer, all they asked for is the normal vectors, aren't these the normal vectors? Why must I add and subtract them like this?:
$13\cdot \begin{pmatrix}1\\ 4\\ 8\end{pmatrix}+9\cdot \begin{pmatrix}3\\ 4\\ 12\end{pmatrix}$
$13\cdot \begin{pmatrix}1\\ 4\\ 8\end{pmatrix}-9\cdot \begin{pmatrix}3\\ 4\\ 12\end{pmatrix}$
b) The solution says that I must choose one component, e.g: $x=2$ and then I substitute it into the equations and complete the simultaneous equation to find the point. Must it be only the x component? And why the value 2? Can it be any value?
So according to the solution, the shared point is  $P(2, 5, -9)$
c) The solution uses the answers from part a and b and gets this $$\begin{pmatrix}10\\ 22\\ 53\end{pmatrix}\cdot \left[\begin{pmatrix}x\\ y\\ z\end{pmatrix}-\begin{pmatrix}2\\ 5\\ -9\end{pmatrix}\right]$$
$$\begin{pmatrix}-7\\ 8\\ -2\end{pmatrix}\cdot \left[\begin{pmatrix}x\\ y\\ z\end{pmatrix}-\begin{pmatrix}2\\ 5\\ -9\end{pmatrix}\right]$$
Is it a general rule to use the normal to find the equation from a shared point?

A:

Answering your questions one by one:
(a)The hidden process going on is to convert the direction ratios of normal to direction cosines and then take their sum, when you do it, the first plane becomes,
$$\frac{x+4y+8z+50}{9}=0$$
$$\frac{3x+4y+12z+82}{13}=0$$
Now, taking the direction ratio of the angle bisectors are the sum and difference of $\pmatrix{\frac{1}{9}\\\frac{4}{9}\\\frac{8}{9}}$ and $\pmatrix{\frac{3}{13}\\\frac{4}{13}\\\frac{12}{13}}$. These are $\frac{1}{9\times 13}$ times the direction ratios you obtain.
(b)Nothing is compulsory. Choose any value of any coordinate you want and solve for other two.
(c)No, is not general rule, but it is the easiest method. You can use any other method you know.

