Distance Formula

The distance formula is used to find the distance between two points in cartesian coordinate system and can be calculated using the distance formula.

Let's say you have two points: A (x_1​, y_1​) and B (x₂​, y₂​).

Distance 'd' between A and B is given by the formula: d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}the

Derivation

Derivation of the Distance Formula is given using Pythagoras Theorem. In the right-angled triangle ABC, we have:

AB² = AC² + BC²

• Distance between points A and C is (x₂ - x₁)
• Distance between points B and C is (y₂ - y₁)

Distance, d is calculated as:

d^2 = {(x_2-x_1)^2 + (y_2-y_1)^2}

Now, taking the square root on both sides,

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Thus, d is the distance between two points.

Distance between Two Points in Polar Co-ordinates

We can also calculate the distance between two points using the polar coordinates.

Let's take two points A (r₁, θ₁) and B (r₂, θ₂) then the distance between them is calculated using the Distance Formula:

AB = \sqrt{r_1^2 + r_2^2 - 2r_1 r_2 \cos(\theta_1 - \theta_2)}

Distance From a Point To a Plane

In 3D we can also calculate the distance between a point and the plane using the distance formula.

Let's take point A (x₁, y₁, z₁) and plane P: ax + by + cz + d = 0 then the distance between the point and the plane is given using the Distance Formula:

d = \frac{|ax_1 + by_1 + cz_1|} { \sqrt{(a^2 + b^2 + c^2)}}

The image below shows the distance between the point and the plane:

Applications

Distance Formula has various applications in Mathematics, Sciences, and others and some of the most important applications of the Distance Formula are:

• The distance of a point from the origin is calculated using the Distance Formula.
• Distance between two points in 2D and 3D planes is calculated using the Distance Formula.
• The magnitude of the vector is calculated using the Distance Formula.
• The magnitude of the complex number is calculated using the Distance formula.

Apart from this distance formula also has some real-life applications which are:

• Distance Formula is used to find the distance between two stars.
• Distance Formula is also used to find the distance between various points in the sea and oceans.
• Distance Formula is used to find the linear distance between two points in the Globe.

Also Check

• 3D distance formula
• Distance of a point from a line
• Distance between two Lines
• Shortest distance between two lines in 3d

Solved Examples

Example 1: Calculate the distance between the points X(5, 15) and Y(4, 14).

Distance between point X and Y is given by distance formula

d = √[( x₂ - x₁ )² + ( y₂ - y₁ )²]

⇒ d = √[( 4 - 5 )² + ( 14 - 15 )²]

⇒ d = √[( 1 )² + ( 1 )²]

⇒ d = √(2)

Distance between X and Y is √2 or 1.41

Example 2: Find the distance between the parallel lines -6x + 20y + 10 = 0 and -6x + 20y + 20 = 0.

General equation of parallel lines is

Ax + By + C₁ = 0 and Ax + By + C₂ = 0, 

Here, 

A = -6, B = 20, C₁ = 10, and C₂ = 20

Applying formula

d = \frac{|c_2-c_1|}{\sqrt{a^2+b^2}}

⇒ d = \frac{|20-10|}{\sqrt{-6^2+20^2}}\\ \frac{|10|}{\sqrt{36+400}}\\ \frac{10}{\sqrt{436}}

⇒ d = 10 / √436

Thus, the distance between two parallel lines is d = 10 / √436

Example 3: Calculate the distance between line 4a + 6b – 26 = 0 from the point (2, –4) using the Distance Formula in maths.

General equation of parallel lines is

A point (x₁, y₁) and a line ax + by + c = 0

Here,

A = 4, B = 6 and C = –26

Applying formula

d = \frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}

⇒ d = \frac{|4\times2+6\times(-4)+(-26)|}{\sqrt{4^2+6^2}}\\ =\frac{8-24-26}{\sqrt{16+36}}\\ =\frac{-42}{\sqrt{52}}

⇒ d = -42 / √52

Thus, the distance between lines and point is d = -42 / √52

Example 4: Calculate the distance between points A(-25, -5) and B(-16, -4) using the Distance Formula in Maths.

Distance between point A and B is given by distance formula

d = √[( x₂ - x₁ )² + ( y₂ - y₁ )²]

⇒ d = √[( (-16) - (-25) )² + ( (-4) - (-5) )²]

⇒ d = √[( 9 )² + ( 1 )²]

⇒ d = √(82)

Distance between A and B is √82 or 9.05

Practice Problems

1. A square has one vertex at (1, 1) and the opposite vertex at (4, 4). Calculate the length of the diagonal.

2. Calculate the distance between the points (-5, -5) and (-1, -1).

3. Find the distance between the points (2, -3) and (-4, 5).

4. Two points, A and B, have coordinates (2, 3) and (10, 7) respectively. A third point, C, lies on the line segment AB and divides it into a ratio of 2:1. Find the coordinates of point C and calculate the distance from C to point A.

5. Calculate the distance between the points (1, 2, 3) and (4, 6, 9).

6. Two vertices of a cube are (0, 0, 0) and (3, 3, 3). Determine the length of the diagonal that connects these two vertices inside the cube.

Answer:-

1. 3√2
2. 4√2
3. 10
4. (8√5)/3
5. √61
6. 3√3