Distance Formula – Examples with Applications

Distance is the space between two places, objects, or things. In geometrical terms, distance is the line between two points, two planes, a point, and a plane. But, why must one know the distance formula? Well, it helps construct new buildings, find locations, route new ways in emergent cases, and much more. Therefore, it is necessary to learn how to find the distance between two points in both three and two dimensions. Then, what is the distance formula? Let’s find out!

Derivation of Distance Formula

According to Pythagoras’ theorem, 

AB² = AC² + BC² 

d² = (x₂-x₁)²+(y₂-y₁)²

Taking the square root on both sides of the equation,

d = √[(x₂-x₁)²+(y₂-y₁)²]

This is referred to as the distance between two points formula. 

Distance Formula Between Two Points in 2D 

The distance formula used in general is the Euclidean distance formula, derived by Euclid’s famous mathematician. Consider two points (x₁, y₁) and (x₂, y₂) in the two-dimension, then the distance between these two points is given by,

d = √[(x₂-x₁)²+(y₂-y₁)²]

Do you know how this distance formula has been derived? This has been derived from the Pythagorean theorem. Consider a right-angled triangle with sides AB and AC and BC be the hypotenuse of the triangle. Then according to the Pythagorean theorem,

BC² = AB² + AC²

d² = (x₂-x₁)²+(y₂-y₁)²

Taking square root on both sides, we get,

d = √[(x₂-x₁)²+(y₂-y₁)²], which is the required distance between two points formula in two dimensions.

Distance Formula Between Two Points in 3D

Do you know that we can also figure out the distance between two points using it in the three dimensions? Consider two points having coordinates (x₁, y₁, z₁) and (x₂, y₂, z₂). Then according to the formula, the distance is given by,

d = √[(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²]

However, the coordinates can increase as the dimensions increase, i.e., in four-dimension, the number of coordinates will be four, and so on.

Distance Formula from a Point to a line in 2D

In the two-dimensional world, one can calculate the distance between a point (x₁, y₁) and a line ax + by + c = 0 using the distance formula geometry. The distance between a point and a line is the perpendicular distance between them. Therefore, the distance is given by,

d = |ax₁+by₁+c|/√a²+b²

Distance Formula from a Point to a line in 3D

Finding distance in the three-dimensional world is not easy. To understand this geometry, one must have a thorough knowledge of the concept. Moreover, the distance formula physics is quite easy to grasp once understood thoroughly. Consider a line whose equation is denoted by x-x₁/a=y-y₁/b=z-z₁/c and a point with coordinates P (x_o, y_o, z_o). Then the distance formula between these is given by,

d = |PQ x s|/|s| , where

• P has the coordinates (x_o, y_o, z_o) and is a point from which distance needs to be calculated through the line.
• Q has the coordinates (x₁, y_1, z₁) on the line whose equation is = x-x₁/a=y-y₁/b=z-z₁/c  .
• PQ = (x₁ – x₀, y₁ – y₀, z₁ – z₀)
• s = <a, b, c> is the direction vector for the line = x-x₁/a=y-y₁/b=z-z₁/c.
• The cross product PQ x s is of s and PQ.

Distance Formula between two Parallel lines

When do we say two lines are parallel to each other? Two lines are parallel when their slopes are the same or are in the same ratio. For example, line 1 is given by 4x + 5y + 8 = 0 and line 2 is given by 4x + 5y + 24 = 0. These two lines are parallel because their slopes are in the same ratio. The slope for line 1 is 8, and the slope for line 2 is 24.

Then what is the distance formula between two parallel lines? The distance between two parallel lines is given by,

d = |C₁-C₂|/√A²+B²

Here, C₁ and C₂ are the slopes for lines 1 and 2, and A and B are the coefficients of x and y, respectively.

Applications of Distance Formula 

Why do we study distance formula physics? Is knowing this geometry important in the real world? It has many applications in the real world. A few of them are listed below.

• It can be used to find the distance between any object, whether in two or three dimensions.
• This is used in complex numbers to represent the plane and its magnitude.
• Moreover, with the help of this, the distance between two planes can be found in three-dimensional or n-dimensional planes.
• It is also used to derive the magnitude formula needed in vectors.
• In geography, It is used to find the distance between two coordinates, two countries, or two far-off places with different topographical conditions.
• Surveyors and engineers use it to prepare maps and adjust the coordinates of new locations.
• The distance between two cities or countries by air is calculated using it.
• Various sailors use this formulae to locate their positions using maps and coordinates in the sea.

Summary of Distance Formulas

 Distance Formula Examples

Example 1: What is the distance between two coordinates A (6, -13) and B (-3, 8)? Solution: From the distance formula, we know the distance between two points is denoted by, d = √[(x-x1)²+(y-y1)²] = √(6-(-13))²+(-3-8)² = √(19)²+(-11)² = 21.954 Example 2: What is the distance between a set of parallel lines -6x + 5y + 2 = 0 and -6x + 5y + 4 = 0? Solution: We know the set of parallel lines is represented by Ax + By + C1 = 0 and Ax + By + C2 = 0. Comparing these equations with the equations given in question, we get, A = -6, B = 5, C1 = 2, C2 = 4 From the distance formula, we have, d = |C1-C2|/√A²+B² Putting the values in the given equation, we get, d = |2-4|/√-6²+5²= 0.256. Example 3: Find the distance between a point (9, 6, 8) and a plane whose equation is 6x + 4y + 2z + 10 = 0. Solution: From the distance formula, we know that the distance between a point (x1, y1, z1) and a plane Ax + By + Cz + d = 0 is given by, d = |Ax1+By1+Cz1+d|/√A²+B²+C² Here, x1= 9, y1 = 6, z1 = 8, A = 6, B = 4, C= 2, d = 10 Putting the values in the above equation, we get, d = |6×9+4×6+2×8+10|/√6²+4²+2² Solving the above equation, we get, d = 14.05.

Frequently Asked Questions

1. What is the Distance Formula in Coordinate Geometry?

Distance can be calculated using the Pythagorean theorem formula. In coordinate geometry, the distance formula is d = √[(x₂-x₁)²+(y₂-y₁)²]

2. What is 2D Distance Formula?

The distance formula calculates the 2D distance between two points. Consider the points P1 and P2, whose coordinates are P1(x_{1 ,} y₁)  

and P2 (x₁ , y₁ )  (x_{2 ,} y₂)

d = |ax₁+by₁ +c|/√a²+b²

3. What is 3D Distance Formula?

3D distance formula is

d = |PQ x s|/|s|

 P has the coordinates(x0, y0, z0) and is a point from which the distance through the line must be calculated.    

 Q is located on the line with the coordinates (x₁, y₁, z₁) and the equation = x-x₁/ay-y₁ /b=z-z₁/c

  PQ = (x₁ – x0, y₁ – y₁, z₁ – z0)

  s = <a, b, c> is the direction vector for the line = x-x₁ /a=y- y₁/b=z-z₁ /c.

  The cross product PQ x s is of s and PQ.

4. How to Derive Distance Formula?

The Pythagorean theorem was used to derive this distance formula. Consider a right-angled triangle with sides AB and AC, with BC as the hypotenuse. The Pythagorean theorem states that

BC² = AB² + AC²

d² = (x₂-x₁)²+(y₂-y₁)²

Taking square root on both sides, we get,

d = √[(x₂-x₁)²+(y₂-y₁)²], which is the two-dimensional formula for the required distance between two points

5. How is the Pythagorean Theorem Related to the Distance Formula?

Pythagorean Theorem They are related to the Distance Formula because they use the same principle and produce a right triangle in both cases.