Dimensional Distance

We often deal with distances in real life. In order to travel from one place to another, we first need to know the distance between two places. We need to perform an analysis of dimensions and calculate the distance between two points while building a house. Such is the significance of the dimensional distance.

The distance between two points in space is the length of the line segment that exists between these two points. The space can be one dimensional, two dimensional, or of higher dimensions. Let’s learn how to calculate the distance between two points in all dimensions.

Dimensional Distance

The distance between two points in one dimension, two dimensions, or any higher dimensions is called the dimensional distance. Usually, the two points are considered to be in the Euclidean space.

The Euclidean space is a space in geometry where the axioms and the rules of the Euclidean geometry are applicable.

The distance between objects that are not points can also be calculated in multiple dimensions. Here, the distance between any two points of those two objects is considered. A method like the Hausdorff distance is used to calculate generally.

The distance of the following objects and points can be calculated with this method.

• Distance between a point and a line, both are lying in the Euclidean space

• Distance between a point and a plane both lying in Euclidean space with three dimensions

• Distance between two lines which lie in three-dimensional Euclidean space

History of the Dimensional Distance

The Euclidean space in which the euclidean distance between any two points in one dimension, two dimensions, or multiple dimensions is calculated is named after Euclid, who was a Greek Mathematician. But even before Euclid came with the Euclidean distance formula, another great Greek Mathematician Pythagoras had worked on it. 

Hence, the Euclidean is also sometimes called the Pythagorean distance. However, there are many other ways to measure the distance between any two points in geometrical spaces other than the Euclidean formula. The Euclidean norms and the rules for the Euclidean space with more than three dimensions were proposed by Augustin-Louis Cauchy.

The Distance Formula

The distance between any two objects in one dimension, two dimensions, or multiple dimensions is generally the smallest distance between two points from the two objects.

Let us take a look at how to calculate this distance between two points in any dimension with formulas one by one.

One Dimension

For two points in one dimension, the distance between those two points on a real line is an absolute value. This absolute value is of the numerical distance between their coordinates. 

Hence, if a and b are considered as two points on a real line, then the distance between them can be calculated with the help of the following formula.

d (a, b) = |a – b|

Since the distance cannot be a negative value, the difference of a and b is enclosed within the modulus.

There is another formula to calculate the distance between two points on a real line.

d (a, b) = √(p – q)2

This formula is a little complicated, and here, we square the difference between the two points and then take the square root of the obtained term.

This works just like the modulus in the above formula.

No difference is made in the positive numbers, but negative numbers get replaced by absolute values.

Two Dimensions

Here the two points which will be calculating the distance exist in two-dimensional spaces.

Suppose there is a point with (a1, a2) as its Cartesian Coordinates.

And there is a point b with (b1, b2) as Cartesian Coordinates, the distance between these points a and b will be calculated by the following formula.

d (a, b) = √(( b1 – a1 )2 + ( b2– a2)2)

We can do this if we apply the Pythagoras theorem to the right angle, which has vertical and horizontal sides, and the distance between points A and B represent the hypotenuse of the right-angle triangle.

Three Dimensions

The formula for calculating the distance between two points in three-dimensional space for higher dimensional spaces is a bit complicated.

Let’s take a look at first to calculate the distance between two points in a three-dimensional space.

Suppose a and b are the two points in a three-dimensional space.

Point a has a1, a2, and a3 as its coordinates in space, while b1, b2, and b3 are the coordinates of point b in space.

d (a, b) = √(( b1 – a1 )2 + ( b2– a2)2 + (b3 – a3)2

First, we square the terms, and then we take their square roots as it makes the negative numbers into their absolute values.

Conclusion

We often delete the distance between two points in real life. These two points can lie in a one-dimensional space, two-dimensional space, three-dimensional space, or even higher dimensions. These spices are called the Euclidean spaces.

The Euclidean formula can be used to find the distance between two points in these spaces.

The Euclidean formula for higher dimensional spaces is

d(a, b) = √(( b1 – a1 )2 + ( b2– a2)2 + (b3 – a3)2

The Euclidean distance is always positive and is symmetric and follows the inequality law of the triangles.