Distance in equation

Distance in physics and mathematics are two different concepts. In physics, it is defined in terms of how much distance a person can travel with the speed, in a particular time interval. While in mathematics, defines the distance between two coordinates on a plane. The distance can be distinguished as the expanse between two objects or points. Here, we define the space between two points as |XY|. If a person moves from one point to another, then the path travelled by the person is referred to as the distance travelled, or the distance between two points. 

X•————————•Y

Overview of Definitions of Distance 

Distance refers to the space between two points. In other words, it tells us how far two points are. The distance between the two points is illustrated with the denotation |XY|. Generally, one measures the distance in two ways, either one wants to measure the physical distance or the other one is to measure the non-physical entities. Non-physical entities include ways like Hamming distance, Lee distance, Levenshtein distance and Jaro-Winkler distance. 

Physical distances further encompass several aspects of it. One can measure the distance travelled, geodesic distance and straight-line distance. Here, the straight-line distance pertains to the quickest apparent route between two points that could be undertaken without any kind of impediments in its way. Apart from that, the geodesic distance is the length of the quickest path, while the object remains on the same plane.

Distance formula 

The distance formula measures and helps us in calculating the space or expanse between two points. However, in a plane, the distance can only be known on the condition that the coordinates of the points are known to us. We can not only measure the distance between two points but even the distance between a line and a point and two lines. 

1)Distance between two points X and Y in a two-dimensional plane

In the simplest ways, these coordinates can either be on the x-axis or the y axis. 

Let’s understand this with an example of two coordinates X and Y where X lies on coordinates (x0,y0) and Y lies on the coordinates (x1,y1). Thus, the distance formula for calculation of the space between these two points is given by: 

            XY = √{(x1-x0)² + (y1-y0)²}

2) Distance between two points X and Y in a three-dimensional plane

Here, for the calculation of distance between two points X and Y in a three-dimensional plane, we take coordinates (x0, y0, z0) for X and Y is situated on the coordinates (x1, y1, z1). 

       XY = √{(x1-x0)² + (y1-y0)² + (z1-z1)²}

3) Distance between a line and a point in a two-dimensional plane

The distance formula for calculating the distance between a line and a point in a two-dimensional plane involves the length of the perpendicular drawn from the point on the line.

Here, we will put up with the equation of the line as px+qy+r=0 and the coordinates of the point M as (x0,y0). Now, to measure the distance between the two, we will be bringing out a perpendicular from the point towards the line, which will touch it at L. Thus, the distance formula for calculating the distance between M and L is: 

                    ML = |px0+qy0+r|

                               √(p²+q²)

4) Distance between two parallel lines in a two-dimensional plane 

Being parallel to each other, the two lines will have the same slope. Hereby, we can use the equations for the lines L0 as px+qy+r1=0 and L1 as px+qy+r2=0. Now, to calculate the distance between two parallel lines in a two-dimensional plane:

            Distance = |  r2– r1 |

                               √(p²+q²)

Applications of the distance formula 

There are numerous applications for the distance formula in mathematics. 

1)One can use this to measure the distance or space between two points. 

2) One can measure distance in any kind of dimensional plane, till n. 

3) The distance formula can even be used to find the magnitude of a vector. 

4) After knowing the geometric coordinates, one can easily find the distance between the two points. 

5) The shortest distance to be travelled can be calculated by the distance formula, thus saving time and money. 

Conclusion

In conclusion, distance refers to the space between two points. However, it should also be noted that the distance calculated from the distance formula is always positive in sign. Therefore, we define the space between two points as |XY|. If a person moves from one point to another, then the path travelled by the person is referred to as the distance travelled, or the distance between two points. Interestingly, we can not only measure the distance between two points but even the distance between a line and a point and two lines.