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 shablack 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.

$\left|\left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)\right|=13$

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?:

$13\cdot \left(\begin{array}{c}1\\ 4\\ 8\end{array}\right)+9\cdot \left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)$

$13\cdot \left(\begin{array}{c}1\\ 4\\ 8\end{array}\right)-9\cdot \left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)$

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 shablack point is $P(2,5,-9)$

c) The solution uses the answers from part a and b and gets this

$\left(\begin{array}{c}10\\ 22\\ 53\end{array}\right)\cdot [\left(\begin{array}{c}x\\ y\\ z\end{array}\right)-\left(\begin{array}{c}2\\ 5\\ -9\end{array}\right)]$

$\left(\begin{array}{c}-7\\ 8\\ -2\end{array}\right)\cdot [\left(\begin{array}{c}x\\ y\\ z\end{array}\right)-\left(\begin{array}{c}2\\ 5\\ -9\end{array}\right)]$

Is it a general rule to use the normal to find the equation from a shablack point?