How to Calculate Distance?

Below, we will calculate distance in different ways such as:

Distance, Speed, and Time Formula
Distance Between Two Points Using Coordinates
Distance Between Two Lines
Distance Between Two Planes

Let’s discuss each case in detail as follows.

Distance, Speed, and Time Formula

One of the fundamental formulas used to calculate distance involves the relationship between distance, speed, and time.

The distance formula is given by:

Distance = Speed × Time

This formula expresses that the distance traveled is equal to the product of the speed at which an object is moving and the time it has been moving at that speed.

Distance Between Two Points Using Coordinates

The formula to calculate the distance between two points using coordinates (x₁, y₁) and (x₂, y₂) is given by:

Distance = √ [(x₂ – x₁)² + (y₂ – y₁)²]

Here, (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. This distance formula is also known as Pythagorean theorem.

Distance Between Two Lines

The distance between two lines can be determined between two parallel or two non-parallel lines.

Distance Formula for Two Parallel Lines

The formula to calculate the distance between two parallel lines when equations are of the form y = mx + c₁ and y = mx + c₂ respectively is given by:

Distance = [Tex]\bold{\frac{|c_1 – c_2|}{\sqrt{1 + m^2}}}[/Tex]

The formula to calculate the distance between two parallel lines when equations are of the form ax + by + c₁ = 0 and ax + by + c₂ = 0 respectively is given by:

Distance = [Tex]\bold{\frac{|c_1 – c_2|}{\sqrt{a^2 + b^2}}}[/Tex]

Distance Formula for Two Non-Parallel Lines

The formula to calculate the distance between two non- parallel lines whose equations are [Tex]\overrightarrow{\rm r_1} = \overrightarrow{\rm a_1}+t\overrightarrow{\rm b_1}[/Tex] and [Tex]\overrightarrow{\rm r_2} = \overrightarrow{\rm a_2}+t\overrightarrow{\rm b_2}[/Tex] respectively is given by:

[Tex]\bold{D = \frac{(\overrightarrow{\rm a_2}-\overrightarrow{\rm a_1}).(\overrightarrow{\rm b_2}-\overrightarrow{\rm b_1})}{(\overrightarrow{\rm b_1}\times \overrightarrow{\rm b_2})}}[/Tex]

Distance Between Two Planes

In three-dimensional space, calculating the distance between two planes involves finding the shortest distance between any two points on the planes. This can be done using vector algebra and the dot product.

The formula to find the distance between two planes is:

[Tex]\text{Distance} = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}[/Tex]

Here, ax + by + cz + d = 0 represents the equation of the plane, and (x₁, y₁, z₁) is a point on the plane.