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The problem asks for the calculation of the distance between two points \( A(2, -1, 7) \) and \( B(-4, 5, 0) \) in a 3D coordinate system.

To find the distance \( AB \) between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in 3D space, we use the distance formula:

\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

For points \( A(2, -1, 7) \) and \( B(-4, 5, 0) \):
- \( x_1 = 2 \), \( y_1 = -1 \), \( z_1 = 7 \)
- \( x_2 = -4 \), \( y_2 = 5 \), \( z_2 = 0 \)

Now, substitute these values into the formula:

\[
AB = \sqrt{(-4 - 2)^2 + (5 - (-1))^2 + (0 - 7)^2}
\]
\[
= \sqrt{(-6)^2 + (6)^2 + (-7)^2}
\]
\[
= \sqrt{36 + 36 + 49}
\]
\[
= \sqrt{121}
\]
\[
= 11
\]

So, the distance \( AB \) is \( 11 \) units.

Would you like further details or have any questions?

Here are some related questions to expand on this concept:
1. What is the formula for finding the midpoint of two points in 3D space?
2. How does the distance formula differ between 2D and 3D space?
3. Can you explain how the Pythagorean theorem is related to the distance formula?
4. How would you find the distance if one of the points was at the origin?
5. How does distance calculation change in non-Euclidean geometries?

**Tip:** Always double-check the sign of each coordinate when using the distance formula, as errors in sign can lead to incorrect results.

মূলবিন্দু O এর সাপেক্ষে দুটি বিন্দু A(2, -1, 7), B(-4, 5, 0) হলে |AB| নির্ণয় কর।

Calculate the distance between two points A(2, -1, 7) and B(-4, 5, 0) in 3D space using the distance formula. Learn key concepts in 3D coordinate geometry and apply the Pythagorean theorem. Suitable for high school students in Grades 9-12.

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