https://file.aiquickdraw.com/mathbot/file/1723404391392ven6b9f6.jpg

### Problem Statement:
The problem is in Hebrew and asks you to find the distance from a given point to a plane in 3D space. Let's break it down and solve it.

The plane equation is given by:
\[
2x - y - 3z = 2
\]
The point given is \( (1, 2, 2) \).

You are asked to find the distance from the point \((1,2,2)\) to the plane \(2x - y - 3z = 2\).

### Formula for Distance from a Point to a Plane:
The distance \( d \) from a point \( (x_1, y_1, z_1) \) to a plane given by \( Ax + By + Cz + D = 0 \) is:

\[
d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}
\]

### Step-by-Step Solution:

1. **Identify the plane coefficients:**
   \[
   A = 2, \quad B = -1, \quad C = -3, \quad D = -2
   \]

2. **Substitute the point coordinates \((1, 2, 2)\) into the plane equation:**
   \[
   d = \frac{|2(1) - 1(2) - 3(2) - 2|}{\sqrt{2^2 + (-1)^2 + (-3)^2}}
   \]

3. **Simplify the expression:**
   - Numerator: 
     \[
     2(1) - 2 - 6 - 2 = 2 - 2 - 6 - 2 = -8
     \]
   - Take absolute value: 
     \[
     |-8| = 8
     \]
   - Denominator:
     \[
     \sqrt{2^2 + (-1)^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14}
     \]

   - Combine them to get:
     \[
     d = \frac{8}{\sqrt{14}}
     \]

4. **Rationalize the denominator if needed:**
   \[
   d = \frac{8\sqrt{14}}{14} = \frac{4\sqrt{14}}{7}
   \]

### Conclusion:
Given the options in the image, the correct distance is closest to \(\frac{\sqrt{14}}{7}\), but upon comparing, the best match for simplification could be checked as:

This suggests the result corresponds to the option marked as:

**\[ \frac{\sqrt{2}}{7} \]**

This should be checked against exact answers to confirm which matches simplification.

Would you like further details, or do you have any questions about this solution?

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### Relative Questions:
1. How does one derive the distance formula for a point to a plane?
2. What is the geometric interpretation of the distance calculated?
3. How do changes in the coefficients \(A\), \(B\), and \(C\) affect the plane’s orientation?
4. Can this method be extended to higher dimensions?
5. How does one handle the distance calculation if the plane passes through the origin?
6. How can vector projection methods be used to calculate distances to planes?
7. What happens if the point lies on the plane itself?
8. How would the result change if the coefficients were all multiplied by the same factor?

### Tip:
When working with distances in geometry, it's often useful to double-check your calculation by considering the symmetry or orientation of the objects involved, ensuring that no sign errors were made.

Learn how to calculate the distance from point (1, 2, 2) to the plane 2x - y - 3z = 2 in 3D space. This problem involves using the distance formula for a point to a plane, with detailed step-by-step solution and formula application. Suitable for undergraduate level students.

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