The distance between the parallel planes
10x + 2y - 2z = 5 and 5x + y - z = 1 is equal to

To find the distance between the parallel planes given by the equations \(10x + 2y - 2z = 5\) and \(5x + y - z = 1\), we first need to express both planes in a standard form.

1. The first plane can be rewritten in the form \(Ax + By + Cz = D\):
   \[
   10x + 2y - 2z - 5 = 0
   \]

2. The second plane:
   \[
   5x + y - z - 1 = 0
   \]

Next, we need to check if the planes are parallel. For two planes \(Ax + By + Cz = D_1\) and \(Ax + By + Cz = D_2\) to be parallel, their normal vectors must be proportional. The normal vector of the first plane is \((10, 2, -2)\) and of the second plane is \((5, 1, -1)\).

Calculating the ratio of their coefficients:
\[
\frac{10}{5} = 2, \quad \frac{2}{1} = 2, \quad \frac{-2}{-1} = 2
\]
Since the ratios are equal, the planes are indeed parallel.

The distance \(d\) between two parallel planes can be calculated using the formula:
\[
d = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}}
\]
where \(D_1\) and \(D_2\) are the constant terms from the plane equations.

Here, we rewrite the second plane to match the first plane's coefficients:
Multiply the entire equation of the second plane by 2:
\[
2(5x + y - z - 1) = 10x + 2y - 2z - 2 = 0
\]
Thus, we have:
- For plane 1: \(D_1 = 5\)
- For plane 2 (after multiplying): \(D_2 = 2\)

Now, plug these values into the distance formula:
\[
d = \frac{|2 - 5|}{\sqrt{10^2 + 2^2 + (-2)^2}} = \frac{|-3|}{\sqrt{100 + 4 + 4}} = \frac{3}{\sqrt{108}} = \frac{3}{6\sqrt{3}} = \frac{1}{2\sqrt{3}} = \frac{\sqrt{3}}{6}
\]

Thus, the distance between the two parallel planes is:
\[
\frac{\sqrt{3}}{6}
\]

If you have any further questions or need more details, feel free to ask! Here are some related questions for further exploration:

1. How do you determine if two planes are parallel?
2. What is the geometric interpretation of the distance between two planes?
3. Can you derive the distance formula for two parallel lines in a 2D plane?
4. How would the distance change if the planes were not parallel?
5. What applications do distances between planes have in real-world scenarios?

**Tip:** Always ensure that the planes are in the correct form before applying formulas for distance or angles.

The distance between the parallel planes 10x + 2y - 2z = 5 and 5x + y - z = 1 is equal to

Explore how to find the distance between the parallel planes defined by 10x + 2y - 2z = 5 and 5x + y - z = 1. This page provides a detailed step-by-step solution, discussing the concepts of parallel planes, normal vectors, and the distance formula, suitable for students in Grades 10-12.

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