Planes Bisecting the Angles Between Two Planes (Cartesian Form)

Understanding the concept of planes bisecting the angles between two planes is crucial in 3D geometry, especially in fields such as engineering, physics, and computer graphics. In Cartesian form, a plane is typically represented by a linear equation of the form:

$$ ax + by + cz + d = 0 $$

where ( a, b, c ) are the direction ratios of the normal to the plane, and ( d ) is the distance from the origin to the plane along the normal.

Bisector Planes of the Angles Between Two Planes

Given two planes with equations:

$$ P_1: a_1x + b_1y + c_1z + d_1 = 0 $$ $$ P_2: a_2x + b_2y + c_2z + d_2 = 0 $$

The bisector planes of the angles between these two planes are the planes that divide the angles formed by ( P_1 ) and ( P_2 ) into two equal parts. There are two such planes: one that bisects the acute angle and another that bisects the obtuse angle between ( P_1 ) and ( P_2 ).

Equations of Bisector Planes

The equations of the bisector planes can be derived using the direction ratios of the normals to ( P_1 ) and ( P_2 ). The direction ratios for ( P_1 ) are ( (a_1, b_1, c_1) ) and for ( P_2 ) are ( (a_2, b_2, c_2) ).

The equations of the bisector planes are given by:

$$ \frac{a_1x + b_1y + c_1z + d_1}{\sqrt{a_1^2 + b_1^2 + c_1^2}} = \pm \frac{a_2x + b_2y + c_2z + d_2}{\sqrt{a_2^2 + b_2^2 + c_2^2}} $$

The positive sign corresponds to the plane bisecting the acute angle, and the negative sign corresponds to the plane bisecting the obtuse angle.

Important Points and Differences

Aspect	Acute Angle Bisector Plane	Obtuse Angle Bisector Plane
Sign in Equation	Positive (+)	Negative (−)
Nature of Angle	Bisects the smaller angle	Bisects the larger angle
Relative Position to Given Planes	Closer to both planes	Further from both planes
Normal Vector	Sum of normals of ( P_1 ) and ( P_2 )	Difference of normals of ( P_1 ) and ( P_2 )

Examples

Example 1: Find the bisector planes of the angles between the planes ( P_1: x + y + z = 1 ) and ( P_2: x + 2y + 3z = 1 ).

Solution:

First, we calculate the magnitude of the normals to ( P_1 ) and ( P_2 ):

For ( P_1 ): ( \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} )

For ( P_2 ): ( \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} )

Now, we can write the equations of the bisector planes:

Acute Angle Bisector Plane:

$$ \frac{x + y + z - 1}{\sqrt{3}} = \frac{x + 2y + 3z - 1}{\sqrt{14}} $$

Obtuse Angle Bisector Plane:

$$ \frac{x + y + z - 1}{\sqrt{3}} = -\frac{x + 2y + 3z - 1}{\sqrt{14}} $$

Example 2: Given the planes ( P_1: 2x - 2y + z = 3 ) and ( P_2: x + y - z = 1 ), find the plane that bisects the acute angle between them.

Solution:

Magnitude of normals:

For ( P_1 ): ( \sqrt{2^2 + (-2)^2 + 1^2} = 3 )

For ( P_2 ): ( \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3} )

Equation of the Acute Angle Bisector Plane:

$$ \frac{2x - 2y + z - 3}{3} = \frac{x + y - z - 1}{\sqrt{3}} $$

By solving the above examples, we can find the exact equations of the bisector planes in Cartesian form. It's important to note that the bisector planes are unique and will always exist as long as the two given planes are not parallel.

In conclusion, understanding the concept of bisector planes between two planes in Cartesian form is essential for solving problems in 3D geometry. The key is to use the direction ratios of the normals to the given planes and apply the formulae correctly to find the equations of the bisector planes.