Mathematically, the shortest distance between two lines can be determined by measuring their perpendicular distances. Calculating the distance between two perpendicular lines can be done using a formula to measure the perpendicular distance between them. Whenever two lines intersect, the shortest distance between them will be zero.

Similarly, the smallest distance between two non-intersecting parallel lines in the same plane is the shortest distance between two points located on both lines.

Distance Between Two Lines

Any two points lying on straight lines are separated by the shortest distance between two lines. Two points on each line are used to measure the distance between two lines. When calculating the distance between two lines, we often have to deal with parallel, intersecting, or skewed lines.

• Perpendicular Lines

An intersection or meeting of two lines at 90° is described as perpendicular. In mathematics, the distance between two parallel lines is equal to the perpendicular distance between any two points on a line.

• Intersecting Lines

When a line crosses another in a plane, it is called an intersecting line. In the case of two intersecting lines, the shortest distance between two lines eventually equals zero.

• Skew Lines

Skew lines are lines that are neither parallel nor intersect each other. The distance between two skew lines is equal to the distance between their perpendiculars.

If no parallel lines and no intersecting lines exist, the distance between them is zero. Two parallel lines that don’t intersect are measured by measuring the length of the perpendicular line segment between them. Similarly, the length perpendicular to both sides of a skew line is the shortest distance between them.

Formula

Using the formula below, you can determine the distance between two parallel lines:

The following is a representation of two parallel lines:

In equation (i), y = mx +c₁

In equation (ii), y = mx + c₂

‘m’ = slope

The formula for the distance is: |c₁-c₂|/<√(a²+b²)

Distance Between Two Parallel Lines

The shortest distance between two lines may split a plane into two points. Parallel lines can be measured by determining the perpendicular distance between any two points on a line. You can determine this distance by following the steps below.

• Lines parallel to one another should be represented by slope-intercept equations
• The slope should not vary between the two lines
• There must be a calculation of each line’s slope and point of interception
• If you substitute all the numbers in the distance formula, you get the distance between two lines

Accordingly, y = mx + c1and y = mx + c

= |c₁-c₂|/√(a²+b²)

Intercepts

There are two types of intercepts in functions, the y-intercept, and the x-intercept. An intercept of a function is a point on the graph of the function where it intersects the axis.

• X Intercept

In equations with linear coefficients, an x-intercept shows the point at which the line crosses the plane’s x-axis. A linear equation that crosses the x-axis will always have a y-coordinate of 0 when it crosses the x-axis. Y-intercepts have zero coordinates, and x-intercepts have zero coordinates.

• Y Intercept

It lies at the intersection point of a graph with the y-axis and is known as the y-intercept. Any point on the y-axis has a 0 as its x-coordinate. This means that the y-intercept’s x-coordinate is 0.

Euclidean Distance Formula

Geometrically, Euclidean distance describes the distance between two points. An object’s distance can be calculated from the length of a segment connecting two points on a plane. Pythagoras’ theorem can be used to calculate Euclidean distance.

Suppose,

On a two-dimensional plane, (x₁, y₁) and (x₂, y₂) are two points. Below is a formula for distance.

d = √[ (x₂ – x₁)² + (y₂ –y₁)²]

Note:

1. A point’s coordinates are (x₁, y₁)
2. The second point’s coordinates are (x₂, y₂)
3. (x₁, y₁) and (x₂, y₂) cannot be separated by the angle d

Conclusion

The shortest distance between two lines can be calculated by measuring their perpendicular distances using a formula. Parallel, intersecting, or skewed lines are three different lines that need to be considered while calculating. The shortest distance between two lines is an important and interesting concept to understand. The solution process would also be simplified and streamlined with a better understanding of the problem.