Distance between two points in three dimensions

 
 
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Plotting points in three dimensions

In this lesson we’ll look at points that are plotted three-dimensionally and how to find the distance between them.

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Points

A point is a representation of a location in space. Its symbol is a single letter, and it’s represented as a dot. When we’re talking about a three-dimensional point, we have three locations: the ???x???-axis, the ???y???-axis and the ???z???-axis. So a point is represented as ???(x,y,z)???. Here are two points ???A??? and ???B??? plotted in three-dimensional space:

 
points in three dimensions
 

Distance formula

We can use the distance formula for three dimensions to find the length of the line segment that connects two points in three-dimensional space.

???d=\sqrt{{{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}}+{{({{z}_{1}}-{{z}_{2}})}^{2}}}???

where ???{{P}_{1}}=({{x}_{1}},{{y}_{1}},{{z}_{1}})??? and ???{{P}_{2}}=({{x}_{2}},{{y}_{2}},{{z}_{2}})???.


Simplifying radicals

Since the distance formula takes the square root of a series of operations, it can be useful to know how to simplify radicals. Remember when you simplify a radical, that the radical has to have no perfect squares remaining under the radical symbol. For example,

???\sqrt{18}=\sqrt{9\cdot 2}=\sqrt{9}\cdot \sqrt{2}=3\sqrt{2}???

 
 

Finding the distance between two points in three-dimensional space


 
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Calculate the distance between the points

Example

Calculate the distance between Points ???B??? and ???C???.

???B=(4,-5,8)???

???C=(1,-3,2)???

We need to use the distance formula.

???d=\sqrt{{{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}}+{{({{z}_{1}}-{{z}_{2}})}^{2}}}???

???d=\sqrt{{{(4-1)}^{2}}+{{(-5-(-3))}^{2}}+{{(8-2)}^{2}}}???

???d=\sqrt{{{(3)}^{2}}+{{(-2)}^{2}}+{{(6)}^{2}}}???

???d=\sqrt{9+4+36}???

???d=\sqrt{49}???

???d=7??? units


Now let’s try one where we’ll need to simplify the radical.


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We can use the distance formula for three dimensions to find the length of the line segment that connects two points in three-dimensional space.

Example

Calculate the distance between Points ???A??? and ???D???.

???A=(3,9,-2)???

???D=(0,-5,4)???

We need to use the distance formula.

???d=\sqrt{{{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}}+{{({{z}_{1}}-{{z}_{2}})}^{2}}}???

???d=\sqrt{{{(3-0)}^{2}}+{{(9-(-5))}^{2}}+{{(-2-4)}^{2}}}???

???d=\sqrt{{{(3)}^{2}}+{{(14)}^{2}}+{{(-6)}^{2}}}???

???d=\sqrt{9+196+36}???

???d=\sqrt{241}\approx15.5??? units

 
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