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Chapter 9: Problem 82

Give the formula for the distance between the points \(\left(x_{1}, y_{1}, z_{1}\right)\) and \(\left(x_{2}, y_{2}, z_{2}\right)\)

Short Answer

Expert verified
The formula for the distance between two points in 3D space \((x_{1}, y_{1}, z_{1})\) and \((x_{2}, y_{2}, z_{2})\) is \[d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}\]

Step by step solution

01

Understanding the Formula

In a three-dimensional space, the formula to calculate the distance between two points (x1, y1, z1) and (x2, y2, z2) is given by \[d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}\]. This formula is derived from the Pythagorean Theorem extended to three dimensions.
02

Breaking Down the Formula

The terms in the formula represent the following: (x2 - x1) is the difference in the x coordinates between the two points, (y2 - y1) is the difference in the y coordinates, and (z2 - z1) is the difference in the z coordinates. Each difference is squared, and the three squared differences are added together.
03

Understanding the Result

The square root of the sum of the squared differences gives the distance. This is based on the premise that the shortest distance between two points in space is a straight line connecting them, and the formula effectively calculates this straight line distance.

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