 Suppose that m · r = a and n · r = b describe two parallel planes. Derive a formula for the distance
between them. Your answer will be in terms of m, n, a, b.

To derive the formula for the distance between two parallel planes described by the equations:

\[
m \cdot r = a \quad \text{and} \quad n \cdot r = b,
\]

where \(m\) and \(n\) are vectors normal to the respective planes and \(r\) is the position vector, we will follow these steps:

### 1. Parallel Planes Condition
For the planes to be parallel, their normal vectors \(m\) and \(n\) must be proportional, meaning there exists a scalar \(k\) such that:

\[
m = k \cdot n.
\]

This simplifies the setup to two parallel planes with normal vectors related by a constant.

### 2. Distance Between Two Parallel Planes
The general formula for the distance \(D\) between two parallel planes is given by:

\[
D = \frac{|d_1 - d_2|}{\| \text{normal vector} \|},
\]

where \(d_1\) and \(d_2\) are the scalar constants from the plane equations, and \(\| \text{normal vector} \|\) is the magnitude of the normal vector to the planes.

### 3. Apply This to Our System
In our case, the equations are:

\[
m \cdot r = a \quad \text{and} \quad n \cdot r = b.
\]

Since the normal vectors \(m\) and \(n\) are proportional (by the condition of parallelism), we can rewrite the distance formula as:

\[
D = \frac{|a - b|}{\| m \|},
\]

where \(\|m\|\) is the magnitude of the normal vector \(m\).

### Final Answer:

\[
\boxed{D = \frac{|a - b|}{\| m \|}}.
\]

This is the formula for the distance between the two parallel planes in terms of \(m\), \(a\), and \(b\).

Would you like a more detailed explanation or any further clarifications? Here are a few related questions:

1. How can you generalize this formula for non-parallel planes?
2. What if the normal vectors were not proportional? Would they still be parallel?
3. How does this formula change if the planes pass through a specific point?
4. How is the dot product used in determining the plane equation?
5. How does this distance formula apply in three-dimensional space?

**Tip:** The magnitude of a vector is crucial in many geometric and physical interpretations. Always check it when dealing with distances.

Suppose that m · r = a and n · r = b describe two parallel planes. Derive a formula for the distance between them. Your answer will be in terms of m, n, a, b.

Understand how to derive the distance between two parallel planes described by m · r = a and n · r = b. This solution explores key concepts in linear algebra and vector mathematics, including the use of the distance formula. Ideal for college-level students studying linear algebra or geometry.

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