Write the equations of the bisector planes of the dihedral…

Write the equations of the bisector planes of the dihedral angle formed by the planes $x=0$ and $z=0$.

Explanation of Method

The bisector planes of the dihedral angles formed by the planes yOz and xOy.

Step 1: Solution

A bisector plane is the set of points equidistant from the two planes: $\frac{|x|}{1}=\frac{|z|}{1}\ \Longrightarrow\ x=\pm z$.

\[x-z=0\qquad\text{and}\qquad x+z=0\]

Both contain the common line $Oy$, are perpendicular to each other, and make $45^\circ$ with each given plane. For example, $\cos\varphi=\tfrac{|\{1,0,0\}\cdot\{1,0,-1\}|}{\sqrt2}=\tfrac1{\sqrt2}$.

$x-z=0$ bisects the angles where $x$ and $z$ have the same sign.
$x+z=0$ bisects the angles where they have opposite signs.

$x-z=0$ $x+z=0$ $45^\circ$ $z=0$ $x=0$ $Oy$

View along the $y$-axis (the common line, perpendicular to the page): the planes $x=0$, $z=0$ appear as lines, and their bisector planes $x=\pm z$ as the diagonals.

Result

\[\boxed{\,x-z=0\qquad\text{and}\qquad x+z=0\,}\]