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What is Distance?

tape measure

A mathematical description of the space between two objects involves some kind of metric, or a way to functionally describe what it means for objects to be "close to one another," or "far away."

The general mathematical case for distance is defined in this way. For any set M, there is a function d: MxM → R, where R stands in for the set of real numbers that satisfies these three conditions.

1.) The distance d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y. In other words, the distance between two points is always positive, and zero is a point, not a distance.

2.) The distance d(x,y) = d(y,x). That is to say the distance relationship is symmetric. The ordering of point we pick should not have an effect on the distance.

3.) The distance satisfies the triangle inequality, or d(x,z) ≤ d(x,y) + d(y,z). All distances are the shortest path between two points.

In mathematics, this distance function is defined as a metric. Along with the set M, this metric function defines a metric space for distance.

For Euclidean geometries, the distance between two points in defined in the following way. Assuming we have a three-dimensional coordinate system with axes x, y, and z, we can geometrically describe the distance between the two points (x1, y1, z1) and (x2, y2, z2) with the function:

d = sqrt((x2-x1)2 + (y2-y1)2 + (z2-z1)2)

The formula above is derived from the Pythagorean theorem. It can be geometrically derived by constructing a right triangle with of its legs on the hypotenuse of another triangle with its other leg orthogonal to the plane that contains the first triangle. The length of the hypotenuse, c, will be equal to the distance between the two objects via the equation c2 = a2 + b2.

This mathematical definition of distance should not be confused with other kinds of distance such as directed distance and displacement (aka distance along a path). A directed distance is a distance with a direction, or a sense (vector quantity). Along a straight line, a directed distance is a vector joining any two points in a Euclidean vector space. Assuming we start at a point A and end at a point B, the directed distance would be the vector joining point A to point B.

Displacements are a special kind of directed distance, useful in mechanics. They are essentially the minimum distance along some kind of path that a particle travels, not the distance between the point where the particle begins and where it ends. For example, a particle could travel in a complete circle and have a very large displacement but have no distance between the point where it began and the point where it ended.

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