Calculate the distance between the point A <5, -4, 0> and point B<-1, -10, -7>

To calculate the distance between two points in a 3-dimensional space, you can use the distance formula:

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

Given the points \( A(5, -4, 0) \) and \( B(-1, -10, -7) \), let's substitute these coordinates into the formula:

\[
\text{Distance} = \sqrt{((-1) - 5)^2 + ((-10) - (-4))^2 + ((-7) - 0)^2}
\]

Now, simplify the expression:

\[
\text{Distance} = \sqrt{(-6)^2 + (-6)^2 + (-7)^2}
\]

\[
\text{Distance} = \sqrt{36 + 36 + 49}
\]

\[
\text{Distance} = \sqrt{121}
\]

\[
\text{Distance} = 11
\]

So, the distance between points \( A \) and \( B \) is 11 units.

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:
1. How do you calculate the distance between two points in 2-dimensional space?
2. What is the distance between the origin and point \( A(5, -4, 0) \)?
3. How would the distance formula change if we were working in a 4-dimensional space?
4. What is the midpoint between points \( A(5, -4, 0) \) and \( B(-1, -10, -7) \)?
5. How does the distance formula relate to the Pythagorean theorem?

**Tip:** Always ensure to double-check your coordinate substitutions when using the distance formula to avoid calculation errors.

Learn how to calculate the distance between two points in 3-dimensional space using the distance formula. Detailed step-by-step solution for points A(5, -4, 0) and B(-1, -10, -7). Suitable for Grades 9-12.

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