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Finding the Distance Between Two Points in a Three Dimensional Space

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we use Pythagoras' theorem and right triangles to develop an equation to find the distance between any two points in a three-dimensional space.

Imagine a math-enabled dolphin named Dolphinius relaxing underwater in a large tank. Dolphinius wants to swim from his present location to another point in this three-dimensional world. Of course, before heading out, our fastidious friend would like to calculate the shortest distance between these points. Any other route would waste precious dolphin energy.

In this lesson, we will develop an equation to find the distance between any two points in 3D space. This information will be of great help to Dolphinius.

A Point in a Three-Dimensional Space

Three dimensions in the Cartesian coordinate system means x, y and z-axes. In this world, the z-axis points straight up. What if we place a point in the x-y plane? For clarity, this point is colored green.


A point in the x-y plane
A_point_in_the_x-y_plane


The place where x, y and z are all zero is called the origin. A line from the origin to the point is labeled c. Along the y-axis is a distance y2. Completing the triangle is x2 which parallels the x-axis. The reason for the ''2'' subscripts is to prepare us for what is coming later. For now, let's focus on the right triangle formed from the lengths x2, y2, and c. Looking down into the x-y plane we see


Triangle in the x-y plane
Triangle_in_the_x-y_plane


In a right triangle, the longest side, labeled c, is called the hypotenuse. From Pythagoras' theorem,


first_triangle_relation


A Second Point in Three-Dimensional Space

By now you and Dolphinius might be getting the idea of locating points by drawing a line from the origin and then looking for a right triangle. Let's place another green point in our space a distance d from the origin:


A second point
A_second_point


Looking towards the origin from some point on the positive x-axis shows another right triangle:


Distance triangle
Distance_triangle


Line c is the same as before and distance d we've already described. What about z2? It's the length of point d in the z direction, making a line parallel to the z-axis. This line segment forms a 90o angle with the x-y plane. From Pythagoras:


second_triangle_relation


Now we are getting somewhere because we can substitute for c:


combining_the_two_triangle_relations


If we take the square root of both sides:


distance_in_terms_of_xyz


d is the distance. In this example, d is the distance from the origin to the second point. The origin can be written as (0, 0, 0). If we include this ''0'' information:


distance_expression_including_origin


Distance Between Two Points in Three-Dimensional Space

Dolphinius, being a clever dolphin, likes results to be as general as possible. Let's say we have two arbitrary points in space and want to know the distance between them.


Distance between two arbitrary points
Distance_between_two_arbitrary_points


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