Angle Between Two Planes (Vector Form)

In three-dimensional geometry, the angle between two planes is a measure of how far the planes are rotated about their line of intersection. This concept is crucial in fields such as physics, engineering, and computer graphics, where understanding the spatial relationship between different surfaces is necessary.

Understanding Planes in Vector Form

Before we delve into the angle between two planes, let's understand how a plane is represented in vector form. A plane in three-dimensional space can be defined by a point and a normal vector. The normal vector is perpendicular to the plane and determines its orientation.

The equation of a plane in vector form is given by:

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

where:

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

Angle Between Two Planes

The angle between two planes is defined as the angle between their normal vectors. If we have two planes with normal vectors $\vec{n}_1$ and $\vec{n}_2$, the cosine of the angle $\theta$ between the planes is given by the dot product of the normal vectors, normalized by the magnitudes of the vectors:

$$ \cos(\theta) = \frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1| |\vec{n}_2|} $$

where $|\vec{n}_1|$ and $|\vec{n}_2|$ are the magnitudes of the normal vectors.

The angle $\theta$ can be found by taking the inverse cosine (arccos) of the above expression:

$$ \theta = \arccos\left(\frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1| |\vec{n}_2|}\right) $$

Important Points and Differences

Aspect	Description
Range of Angle	The angle between two planes ranges from $0^\circ$ to $90^\circ$.
Parallel Planes	If two planes are parallel, their normal vectors are parallel, and the angle between them is $0^\circ$.
Perpendicular Planes	If two planes are perpendicular, their normal vectors are orthogonal, and the angle between them is $90^\circ$.
Acute and Obtuse Angles	If the dot product of the normal vectors is positive, the angle is acute ($< 90^\circ$). If it is negative, the angle is obtuse ($> 90^\circ$).

Formulas

• Dot Product of Normal Vectors: $\vec{n}1 \cdot \vec{n}_2 = n{1x}n_{2x} + n_{1y}n_{2y} + n_{1z}n_{2z}$
• Magnitude of a Vector: $|\vec{n}| = \sqrt{n_x^2 + n_y^2 + n_z^2}$
• Angle Between Planes: $\theta = \arccos\left(\frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1| |\vec{n}_2|}\right)$

Examples

Example 1: Parallel Planes

Consider two planes with normal vectors $\vec{n}_1 = \langle 2, 3, 4 \rangle$ and $\vec{n}_2 = \langle 4, 6, 8 \rangle$. Since $\vec{n}_2$ is a scalar multiple of $\vec{n}_1$, the planes are parallel, and the angle between them is $0^\circ$.

Example 2: Perpendicular Planes

Let's take two planes with normal vectors $\vec{n}_1 = \langle 1, 0, 0 \rangle$ and $\vec{n}_2 = \langle 0, 1, 0 \rangle$. The dot product $\vec{n}_1 \cdot \vec{n}_2 = (1)(0) + (0)(1) + (0)(0) = 0$, indicating that the planes are perpendicular, and the angle between them is $90^\circ$.

Example 3: Acute Angle Between Planes

Suppose we have two planes with normal vectors $\vec{n}_1 = \langle 1, 2, 3 \rangle$ and $\vec{n}_2 = \langle 4, -1, 2 \rangle$. To find the angle between them:

1. Calculate the dot product: $\vec{n}_1 \cdot \vec{n}_2 = (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8$.
2. Find the magnitudes: $|\vec{n}_1| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}$ and $|\vec{n}_2| = \sqrt{4^2 + (-1)^2 + 2^2} = \sqrt{21}$.
3. Compute the angle: $\theta = \arccos\left(\frac{8}{\sqrt{14}\sqrt{21}}\right)$.

By calculating the arccosine, we find the angle $\theta$ to be acute.

Understanding the angle between two planes in vector form is essential for solving problems in 3D geometry and can be applied to various practical scenarios. The key is to remember that the angle is determined by the relationship between the normal vectors of the planes.