Shortest distance between two parallel lines in vector + cartesian form

The shortest distance between two parallel lines can be determined using both vector and Cartesian forms. Let's break it down.

Understanding the Lines

Consider two parallel lines represented in vector form as:

• Line 1: **r1 = a + t * b**
• Line 2: **r2 = c + s * b**

Here, **a** and **c** are position vectors of points on each line, **b** is the direction vector (the same for both lines), and **t** and **s** are parameters.

Distance Calculation in Vector Form

The shortest distance (D) between the two lines can be calculated using the formula:

D = |(c - a) · n| / |n|

In this formula, **n** is a vector perpendicular to the direction vector **b**. You can find **n** by taking the cross product of **b** with any vector that is not parallel to it.

Cartesian Form Approach

If the lines are given in Cartesian form as:

• Line 1: **y = mx + c1**
• Line 2: **y = mx + c2**

Here, **m** is the slope, and **c1** and **c2** are the y-intercepts. The distance between these two lines can be calculated using the formula:

D = |c2 - c1| / √(1 + m²)

Summary

In both forms, the key to finding the shortest distance lies in understanding the relationship between the lines and using the appropriate formulas. Whether you use vectors or Cartesian equations, the principles remain consistent.