Distance between two points in three dimensions
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.
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:
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
Take the course
Want to learn more about Geoemtry? I have a step-by-step course for that. :)
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.
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