Planes Passing Through the Intersection of Two Planes (Vector Form)

In 3D geometry, understanding the concept of planes and their intersections is crucial. When two planes intersect, they do so along a line. A third plane can pass through this line of intersection, and this concept is often explored in vector form in mathematics. Let's delve into this topic in detail.

Understanding Planes in Vector Form

A plane in 3D 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 we have two planes, Plane 1 and Plane 2, their equations can be written as:

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

Intersection of Two Planes

The intersection of Plane 1 and Plane 2 is a line. This line can be found by solving the two plane equations simultaneously. However, we are interested in a third plane that passes through this line of intersection.

Plane Passing Through the Intersection of Two Planes

The equation of a plane passing through the intersection of two other planes can be written as a linear combination of the equations of the two planes:

$$ \vec{r} \cdot (\lambda \vec{n_1} + \mu \vec{n_2}) = \lambda d_1 + \mu d_2 $$

where $\lambda$ and $\mu$ are real numbers, not both zero.

This equation represents a family of planes that pass through the line of intersection of Plane 1 and Plane 2.

Table of Differences and Important Points

Feature	Plane 1	Plane 2	Plane Passing Through Intersection
Normal Vector	$\vec{n_1}$	$\vec{n_2}$	$\lambda \vec{n_1} + \mu \vec{n_2}$
Distance from Origin	$d_1$	$d_2$	$\lambda d_1 + \mu d_2$
Equation	$\vec{r} \cdot \vec{n_1} = d_1$	$\vec{r} \cdot \vec{n_2} = d_2$	$\vec{r} \cdot (\lambda \vec{n_1} + \mu \vec{n_2}) = \lambda d_1 + \mu d_2$
Represents	A single plane	A single plane	A family of planes

Formulas

The general equation of a plane passing through the intersection of two planes is:

$$ \vec{r} \cdot (\lambda \vec{n_1} + \mu \vec{n_2}) = \lambda d_1 + \mu d_2 $$

Examples

Let's consider two planes with the following equations:

$$ \vec{r} \cdot \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} = 6 \quad \text{(Plane 1)} $$ $$ \vec{r} \cdot \begin{bmatrix} 2 \ -1 \ 4 \end{bmatrix} = 8 \quad \text{(Plane 2)} $$

Example 1: Finding a Plane Passing Through the Intersection

To find a plane that passes through the intersection of Plane 1 and Plane 2, we choose values for $\lambda$ and $\mu$. Let's take $\lambda = 2$ and $\mu = -3$:

$$ \vec{r} \cdot (2 \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} - 3 \begin{bmatrix} 2 \ -1 \ 4 \end{bmatrix}) = 2(6) - 3(8) $$ $$ \vec{r} \cdot \begin{bmatrix} -4 \ 7 \ -6 \end{bmatrix} = -12 $$

This gives us the equation of a plane passing through the intersection of Plane 1 and Plane 2.

Example 2: Verifying the Family of Planes

Let's verify that another choice of $\lambda$ and $\mu$ gives us a different plane that still passes through the line of intersection. Take $\lambda = 1$ and $\mu = 1$:

$$ \vec{r} \cdot (\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} + \begin{bmatrix} 2 \ -1 \ 4 \end{bmatrix}) = 6 + 8 $$ $$ \vec{r} \cdot \begin{bmatrix} 3 \ 1 \ 7 \end{bmatrix} = 14 $$

This is another plane equation, and it also passes through the intersection of Plane 1 and Plane 2.

Conclusion

The concept of a plane passing through the intersection of two planes is a fundamental aspect of 3D geometry. By understanding the vector form of plane equations and the method of combining them, we can find the equation of a family of planes that share a common line of intersection. This topic is particularly important in fields such as computer graphics, engineering, and physics, where spatial relationships are key.