 # Distance Formula

As the name implies, any distance formula calculates the distance (the length of the line segment). The length of the line section connecting 2 places, as an example, is that the distance between them. The Pythagoras theorem is employed to get the space formula between 2 points in a very two-dimensional plane, which can even be extended to seek out the space between 2 points in a very three-dimensional plane. In coordinate geometry, there exist various distance formulas.

1. In a 2-dimensional plane, the distance between the given 2  points
2. The distance between two points in a three-dimensional plane.
3. The distance between a point and a line in 2Dimension.
4. The distance between two parallel lines in 2Dimension.
5. The distance between a point and a line in 3Dimension.
6. The shortest distance between the skew lines.
7. The total distance between any point and a plane.
8. Distance between any 2 parallel lines.

In the following sections, we’ll go over all of these distance formulas in detail, along with some solved examples and practice questions.

What is Distance Formula:

In coordinate geometry, there’s a list of distance formulas that are used to find the distance between the given two points, the distance between the given two parallel lines, the distance between the given two parallel planes, etc. All of the distance formulas are listed below, and we’ll go through each one separately in the sections that follow.

## How to Calculate Distance Between Two Points using Distance Formulae:

We will see the distance between two points in a two-dimensional plane and three-dimensional space. Both distance formulas are derived by using the Pythagoras Theorem.

### Distance Between Two Points in 2D

The distance formula that is employed to seek out the gap between 2 points in an exceedingly two-dimensional plane is additionally referred to as the euclidean distance formula. To derive the formula, let us consider two points in 2D planes A(x1,y1), and B(x2,y2). Suppose that ‘d’ is the distance between A and B.

## Derivation of Distance Formula

By the Pythagoras theorem,

AB2 = AC2 + BC2

d2 = (x²– x1)2 + (y²– y1)2

Taking the square root on both sides,

d = √[(x² – x1)2 + (y² – y1)2]

This is called the distance between two points formula.

### Distance Between Two Points in 3D

To find the distance formula for 2 points in the 3-D plane, let us consider two points in a three-dimensional plane A (x1,y1,z1), and B(x2,y2,y3). Suppose ‘d’ be the distance between P and Q. By applying the same logic (as explained in the previous section) of finding the distance between two points in 2D, the distance between two points in 3D,

## Calculating the Distance From a Point To a Line using the Distance Formula

Now we will learn the distance formula for the distance from a point to a line in 2-D and 3-D. Both formulas do not look similar.

Distance From a degree To a Line in 2d

The distance formula to calculate the space from a degree to a line is that the length of the perpendicular line section that’s drawn from the purpose to the line. Let us consider a line L in a two-dimensional plane with the equation ax + by + c =0 and consider a point P (x1,y1). Then the distance (d) from P to L is, d = |ax1+by1+c|

__________

√a²+b²

## Calculating Distance Between Two Lines using Distance Formulae:

### Distance Between Two Parallel Lines in 2D

Any 2 parallel lines will have same slopes. So we can assume the two parallel lines to be L1: ax + by + c1 = 0 and L2: ax + by + c2 = 0. Then the distance (d) between L1 and L2 is: d=|c2−c1|a2+b2) to a plane ax + by + cz + d = 0 is given by:

## How to Calculate Distance Between Two Parallel Planes using Distance Formula

We know that the traditional vectors of 2 parallel planes are either equal or in proportion.Thus, to find the distance formula between two parallel planes, we can consider the equations of two parallel planes to be ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0. Total distance between two parallel planes is,

d =  |d2−d1|

________

√a²+b²+c²

## Applications of Distance Formula

The distance formula has numerous applications in other areas of mathematics and also in many real-life situations. Some of the uses of the distance formula are as follows.

• The distance of any purpose from the origin may be calculated victimization the space formula.
• The number is depicted within the arg-and plane, and therefore the formula to seek out the magnitude of a fancy variety has been derived from the space formula.
• The distance formula conjointly also can be accustomed to realize the space between 2 points in three-dimensional and also in n-dimensional planes.
• The distance formula may be accustomed to derive the magnitude formula, to seek out the magnitude of a vector.
• The distance between 2 points in a very ocean may be found by distinguishing the geographic coordinates of the 2 points so applying the space formula.
• The distance between 2 cities for the aim of locomoting air is the shortest distance and is calculated victimization the space formula.

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