The equation of the plane bisecting the acute angle between the planes $$x - y + z - 1 = 0$$    and $$x + y + z = 2$$   is :

Solution :
Rewriting the equations
$$\eqalign{ & - x + y - z + 1 = 0 \cr & - x - y - z + 2 = 0 \cr & {\text{Here, }}{d_1} = 1 > 0,\,{d_2} = 2 > 0{\text{ and}} \cr & {a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = \left( { - 1} \right)\left( { - 1} \right) + 1\left( { - 1} \right) + \left( { - 1} \right)\left( { - 1} \right) = 1 > 0 \cr} $$
$$\therefore $$  the bisector of the angle containing the origin is not the bisector of the acute angle.
$$\therefore $$  the required bisector has the equation $$\frac{{ - x + y - z + 1}}{{\sqrt {{{\left( { - 1} \right)}^2} + {1^2} + {{\left( { - 1} \right)}^2}} }} = - \frac{{ - x - y - z + 2}}{{\sqrt {{{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2}} }}.$$