Plane Parallel to a Given Plane (Vector Form)

In three-dimensional geometry, understanding the concept of parallel planes is crucial. Two planes are said to be parallel if they do not intersect at any point, no matter how far they are extended. This concept can be expressed in vector form, which provides a powerful tool for solving geometric problems involving planes.

Equation of a Plane in Vector Form

Before discussing parallel planes, let's recall the general equation of a plane in vector form. A plane can be defined by a point ( \vec{A} ) (with position vector ( \vec{a} )) and a normal vector ( \vec{n} ) which is perpendicular to the plane. The equation of the plane is given by:

[ (\vec{r} - \vec{a}) \cdot \vec{n} = 0 ]

where ( \vec{r} ) is the position vector of any point ( P ) on the plane.

Condition for Parallel Planes

Two planes are parallel if their normal vectors are parallel (or equivalent, collinear). This means that the normal vectors are scalar multiples of each other. If we have two planes with equations:

[ (\vec{r} - \vec{a}) \cdot \vec{n} = 0 \quad \text{(Plane 1)} ] [ (\vec{r} - \vec{b}) \cdot \vec{m} = 0 \quad \text{(Plane 2)} ]

Then, Plane 1 is parallel to Plane 2 if and only if:

[ \vec{n} = k\vec{m} ]

where ( k ) is a non-zero scalar.

Table of Differences and Important Points

Aspect	Plane 1 (Given Plane)	Plane 2 (Parallel Plane)
Normal Vector	( \vec{n} )	( \vec{m} )
Condition for Parallel	N/A	( \vec{m} = k\vec{n} )
Point on Plane	( \vec{a} )	( \vec{b} )
Equation of Plane	( (\vec{r} - \vec{a}) \cdot \vec{n} = 0 )	( (\vec{r} - \vec{b}) \cdot \vec{m} = 0 )
Relationship	Reference Plane	Must have a normal vector proportional to ( \vec{n} )

Formulas

Given the equation of Plane 1:

[ (\vec{r} - \vec{a}) \cdot \vec{n} = 0 ]

The equation of any Plane 2 parallel to Plane 1 can be written as:

[ (\vec{r} - \vec{b}) \cdot \vec{n} = 0 ]

where ( \vec{b} ) is the position vector of any point on Plane 2.

Examples

Example 1: Finding a Parallel Plane

Given the plane with equation ( (\vec{r} - \vec{a}) \cdot \vec{n} = 0 ), where ( \vec{a} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} ) and ( \vec{n} = \begin{bmatrix} 4 \ -5 \ 6 \end{bmatrix} ), find the equation of a plane parallel to this plane that passes through the point with position vector ( \vec{b} = \begin{bmatrix} 7 \ -8 \ 9 \end{bmatrix} ).

Solution:

Since the planes are parallel, they share the same normal vector ( \vec{n} ). Therefore, the equation of the parallel plane is:

[ (\vec{r} - \vec{b}) \cdot \vec{n} = 0 ]

Substituting ( \vec{b} ) and ( \vec{n} ) into the equation, we get:

[ (\vec{r} - \begin{bmatrix} 7 \ -8 \ 9 \end{bmatrix}) \cdot \begin{bmatrix} 4 \ -5 \ 6 \end{bmatrix} = 0 ]

This is the equation of the plane parallel to the given plane that passes through the point with position vector ( \vec{b} ).

Example 2: Verifying Parallel Planes

Verify if the planes with equations ( (\vec{r} - \vec{a}) \cdot \begin{bmatrix} 2 \ 3 \ 4 \end{bmatrix} = 0 ) and ( (\vec{r} - \vec{b}) \cdot \begin{bmatrix} -4 \ -6 \ -8 \end{bmatrix} = 0 ) are parallel.

Solution:

To verify if the planes are parallel, we need to check if their normal vectors are scalar multiples of each other. We have:

[ \vec{n}_1 = \begin{bmatrix} 2 \ 3 \ 4 \end{bmatrix} \quad \text{and} \quad \vec{n}_2 = \begin{bmatrix} -4 \ -6 \ -8 \end{bmatrix} ]

We can see that ( \vec{n}_2 = -2\vec{n}_1 ), which means that the normal vectors are parallel, and hence, the planes are parallel.

By understanding the vector form of the equation of a plane and the condition for parallel planes, we can solve various problems related to the geometry of planes in three-dimensional space.