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The Distance Formula in 3 Dimensions

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The Distance Formula in 3 Dimensions

Study Guide

Key Definition

The distance between two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in 3D space is calculated using: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Important Notes

The formula extends the 2D distance formula by adding the z-dimension.
3D coordinates include $x$, $y$, and $z$ components.
Distance in 3D space is always non-negative.
The formula uses the Pythagorean theorem in three dimensions.
The units of distance depend on the units of the coordinates.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

The distance formula in 3D is derived from the Pythagorean theorem, extending it by considering the third dimension, $z$.

Remember

To find the distance in 3D, always include the $(z_2 - z_1)^2$ term in your calculations.

Quick Reference

3D Distance Formula:$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Understanding The Distance Formula in 3 Dimensions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The 3D distance formula is similar to the 2D version but includes a $z$ component: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the distance between points $(1, 2, 3)$ and $(4, 5, 6)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Suppose you are designing a 3D video game and need to calculate the distance between two objects located at $(7, 8, 9)$ and $(10, 11, 12)$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Imagine you are a drone pilot and need to calculate the flight path between two points at $(1, 4, 7)$ and $(2, 5, 8)$.

Challenge Quiz

Single Choice Quiz

Advanced

Calculate the distance in 3D space between $(-1, -2, -3)$ and $(4, 5, 6)$.

Recap

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Review key concepts and takeaways