How to find the distance between points in three dimensions
The distance formula in three dimensions
Given two points ???A??? and ???B??? in three-dimensional space,
???A(x_1,y_1,z_1)???
???B(x_2,y_2,z_2)???
we can calculate the distance between them using the distance formula.
???D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}???
It doesn’t matter which point is ???A??? and which point is ???B???. The fact that we square the differences inside the square root means that all of our values will be positive, which means we’ll get a positive value for the distance between the points.
Using the distance formula to calculate the distance between two points in 3D space
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Applying the distance formula to identify the location of points and say which point is closer to a plane
Example
Use the distance formula to
Find the distance between ???(0,1,3)??? and ???(-1,4,5)???.
Say which of ???(0,1,3)??? and ???(-1,4,5)??? lies in the ???yz???-plane.
Say which of ???(0,1,3)??? and ???(-1,4,5)??? is closer to the ???xy???-plane.
For the first part of the question, we’ll use the distance formula to calculate the distance between the points.
???D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}???
???D=\sqrt{(-1-0)^2+(4-1)^2+(5-3)^2}???
???D=\sqrt{1+9+4}???
???D=\sqrt{14}???
For the second part of the question, we need to realize that in order for a point to be in the ???yz???-plane, its ???x???-coordinate must be ???0???. With that in mind, we can say that ???(0,1,3)??? lies on the ???yz???-plane, and that ???(-1,4,5)??? does not lie in the ???yz???-plane.
For the third part of the question, we need to realize that the ???z???-value of the coordinate point will tell us how far the point is from the ???xy???-plane. So if we just take the absolute value of the ???z???-coordinate for each of our points, we’ll be able to say which one is closer.
Point ???(0,1,3)??? has ???|z|=|3|=3???
Point ???(-1,4,5)??? has ???|z|=|5|=5???
Since the absolute value of ???z??? in the point ???(0,1,3)??? is less than the absolute value of ???z??? in the point ???(-1,4,5)???, we can say that ???(0,1,3)??? is closer to the ???xy???-plane.