Planes Bisecting the Angles Between Two Planes (Vector Form)

In three-dimensional geometry, the concept of planes bisecting the angles between two planes is an important topic. When two planes intersect, they form four angles. A plane that bisects any of these angles is called a bisecting plane. There are two types of bisecting planes: one that bisects the acute angles and one that bisects the obtuse angles between the two planes.

Understanding the Vector Form of Planes

Before diving into the bisecting planes, let's understand the vector form of a plane. A plane in three-dimensional space can be represented in vector form as:

$$ \vec{r} \cdot \vec{n} = d $$

where:

• $\vec{r}$ is the position vector of any point on the plane,
• $\vec{n}$ is the normal vector to the plane, and
• $d$ is the perpendicular distance from the origin to the plane.

If two planes are given by the equations:

$$ \vec{r} \cdot \vec{n_1} = d_1 \quad \text{(Plane 1)} $$ $$ \vec{r} \cdot \vec{n_2} = d_2 \quad \text{(Plane 2)} $$

then the angle between these two planes is given by the angle between their normal vectors $\vec{n_1}$ and $\vec{n_2}$.

Angle Between Two Planes

The cosine of the angle $\theta$ between two planes is given by the dot product of their normal vectors:

$$ \cos \theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}| |\vec{n_2}|} $$

Bisecting Planes

The bisecting planes of the angles between two planes will have normal vectors that are either the sum or the difference of the normal vectors of the given planes.

Acute Angle Bisector

The normal vector to the plane bisecting the acute angle between two planes is given by:

$$ \vec{n_{acute}} = \frac{\vec{n_1}}{|\vec{n_1}|} + \frac{\vec{n_2}}{|\vec{n_2}|} $$

Obtuse Angle Bisector

The normal vector to the plane bisecting the obtuse angle between two planes is given by:

$$ \vec{n_{obtuse}} = \frac{\vec{n_1}}{|\vec{n_1}|} - \frac{\vec{n_2}}{|\vec{n_2}|} $$

Equations of Bisecting Planes

The equations of the bisecting planes can be written as:

$$ \vec{r} \cdot \vec{n_{acute}} = d_{acute} $$ $$ \vec{r} \cdot \vec{n_{obtuse}} = d_{obtuse} $$

where $d_{acute}$ and $d_{obtuse}$ are the distances from the origin to the acute and obtuse angle bisecting planes, respectively.

Table of Differences and Important Points

Aspect	Acute Angle Bisector	Obtuse Angle Bisector
Normal Vector	$\vec{n_{acute}} = \frac{\vec{n_1}}{	\vec{n_1}
Angle Bisected	Acute Angle	Obtuse Angle
Direction	Towards smaller angle	Towards larger angle
Equation Form	$\vec{r} \cdot \vec{n_{acute}} = d_{acute}$	$\vec{r} \cdot \vec{n_{obtuse}} = d_{obtuse}$

Examples

Example 1: Find the Bisecting Planes

Given two planes with normal vectors $\vec{n_1} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{n_2} = 4\hat{i} + 5\hat{j} + \hat{k}$, find the equations of the planes bisecting the angles between them.

Solution:

First, normalize the normal vectors:

$$ |\vec{n_1}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} $$ $$ |\vec{n_2}| = \sqrt{4^2 + 5^2 + 1^2} = \sqrt{42} $$

Now, find the normal vectors for the bisecting planes:

$$ \vec{n_{acute}} = \frac{\vec{n_1}}{\sqrt{14}} + \frac{\vec{n_2}}{\sqrt{42}} $$ $$ \vec{n_{obtuse}} = \frac{\vec{n_1}}{\sqrt{14}} - \frac{\vec{n_2}}{\sqrt{42}} $$

Assuming $d_{acute}$ and $d_{obtuse}$ are known or can be calculated based on additional information, the equations of the bisecting planes can be written using the above normal vectors.

Example 2: Angle Between Planes

Given two planes with normal vectors $\vec{n_1} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{n_2} = -\hat{i} + \hat{j} + \hat{k}$, find the angle between the planes.

Solution:

Calculate the dot product and magnitudes:

$$ \vec{n_1} \cdot \vec{n_2} = (-1)(1) + (1)(1) + (1)(1) = 1 $$ $$ |\vec{n_1}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} $$ $$ |\vec{n_2}| = \sqrt{(-1)^2 + 1^2 + 1^2} = \sqrt{3} $$

Now, find the cosine of the angle:

$$ \cos \theta = \frac{1}{\sqrt{3} \sqrt{3}} = \frac{1}{3} $$

Therefore, the angle $\theta$ between the planes is $\cos^{-1}\left(\frac{1}{3}\right)$.

Understanding the vector form of planes and the concept of bisecting planes is crucial for solving problems related to the geometry of three-dimensional space. The above examples illustrate how to apply these concepts to find bisecting planes and angles between planes.