Equation of a Plane Bisecting the Angle Between Two Planes
Given planes are
\({{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0\) . . . (1)
\({{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0\) . . . (2)
Let P (x, y, z) be a point on the plane bisecting the angle between (1) and (2).
Let PL and PM be the length to the perpendiculars from P to planes (i) and (2).
Therefore, PL = PM
\(\left| \frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}} \right|=\left| \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}} \right|\),
\(\frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}}=\pm \ \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}}\).
This is the equation of the plane bisecting between Plane (1) and (2)
Vector Form: The equation of the plane bisecting the angle between planes \(\vec{r}.{{\vec{n}}_{1}}={{d}_{1}}\) and \(\vec{r}.{{\vec{n}}_{2}}={{d}_{2}}\) is \(\left| \frac{\vec{r}.{{{\vec{n}}}_{1}}-{{d}_{1}}}{{{{\vec{n}}}_{1}}} \right|=\left| \frac{\vec{r}.{{{\vec{n}}}_{2}}-{{d}_{2}}}{{{{\vec{n}}}_{2}}} \right|\).
Example: Find the equation of the bisectors of the angle between the plane 2x – y + 2z + 3 = 0 and 3x – 2y + 6z + 8 = 0.
Solution: The given planes are 2x – y + 2z + 3 = 0 and 3x – 2y + 6z + 8 = 0.
\(\frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}}=-\ \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}}\),
\(\frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}}=+\ \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}}\),
\(\frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}}=\pm \ \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}}\),
\(\frac{2x-y+2z+3}{\sqrt{{{\left( 2 \right)}^{2}}+{{\left( -1 \right)}^{2}}+{{\left( 2 \right)}^{2}}}}=\pm \ \frac{3x-2y+6z+8}{\sqrt{{{\left( 3 \right)}^{2}}+{{\left( -2 \right)}^{2}}+{{\left( 6 \right)}^{2}}}}\),
\(\frac{2x-y+2z+3}{\sqrt{4+1+4}}=\pm \ \frac{3x-2y+6z+8}{\sqrt{9+4+36}}\),
\(\frac{2x-y+2z+3}{\sqrt{9}}=\pm \ \frac{3x-2y+6z+8}{\sqrt{49}}\),
\(\frac{2x-y+2z+3}{3}=\pm \ \frac{3x-2y+6z+8}{7}\),
\(7\left( 2x-y+2z+3 \right)=\pm 3\left( 3x-2y+6z+8 \right)\).