The shortest distance between a line and a point is achieved when?

A line is drawn at 90 degrees to the given line from the given point

A line is drawn at 180 degrees to the given line from the given point

A line is drawn at 60 degrees to the given line from the given point

A line is drawn at 270 degrees to the given line from the given point

Shortest Distance between Two Lines - Definition, Formula, Proof and Examples

The distance between two parallel lines is used to determine how far apart the lines are. This can be done by measuring the perpendicular distance between them. We may derive a formula using this approach and use this formula directly to find the shortest distance between two parallel lines. For two non-intersecting lines lying in the same plane, the shortest distance is the distance that is shortest of all the distances between two points lying on both lines. In this article, we will study the shortest distance between two lines in detail.

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Table of Contents

Distance between Two Straight Lines
Shortest Distance between Two Parallel Lines – Formula and Proof
Shortest Distance between Skew Lines
Solved Problems

Distance between two Straight Lines

The distance between two straight lines in a plane is the minimum distance between any two points lying on the lines. In geometry, we often deal with different sets of lines, such as parallel lines, intersecting lines or skew lines.

Distance between Two Parallel Lines	The distance is the perpendicular distance from any point on one line to the other line.
Distance between Two Intersecting Lines	The shortest distance between such lines is eventually zero.
Distance between Two Skew Lines	The distance is equal to the length of the perpendicular distance between the lines.

Straight Lines

3D Geometry

Distance Formula

Shortest Distance between Two Parallel Lines

Formula to find the distance between two parallel lines:

Consider two parallel lines that are represented in the following form:

y = mx + c₁ …(i)

y = mx + c₂ ….(ii)

Where m = slope of the line

Then, the formula for the shortest distance can be written as under:

\(\begin{array}{l}d = \frac{|c_2 – c_1|}{\sqrt{1+m^2}}\end{array} \)

If the equations of two parallel lines are expressed in the following way,

ax + by + d₁ = 0

ax + by + d₂ = 0

Then, there is a small change in the formula.

\(\begin{array}{l}d=\frac{\left | d_{2} -d_{1}\right |}{\sqrt{a^{2}+b^{2}}}\end{array} \)

Remark: The perpendicular distance between parallel lines is always a constant, so we can pick any point to measure the distance.

Proof

Consider two parallel lines given by

y = mx + c₁ ..(i)

y = mx + c₂ ..(ii)

Shortest Distance between Two Lines

Here, line (i) intersects the x-axis at A. So, y = 0 at that point.

We can write (i) as 0 = mx + c₁

So, mx = -c₁

x = -c₁/m

Point A will be (-c₁/m, 0).

The perpendicular distance from A to line (ii) is the distance between lines (i) and (ii).

The equation of line (ii) can be written as mx – y + c₂ = 0

Comparing with general equation Ax+ By + C = 0

We get A = m, B = -1, C = c₂

Here, (x₁, y₁) = (-c₁/m, 0)

The distance d = |(Ax₁ + By₁+C)/√(A² + B²)|

= |(m(-c₁/m) + -1(0) + c₂)/√(m² + 1)|

= |(-c₁+ 0 + c₂)/√(m² + 1)|

= |(c₂-c₁)/√(1 + m²)|

In Vector Form:

\(\begin{array}{l}\text{If}\ \vec{r}=\vec{a_1} + \lambda \vec{b}\ \text{and}\ \vec{r}=\vec{a_2} + \mu \vec{b}\end{array} \)

Then,

\(\begin{array}{l}d = |\frac{\vec{b} \times (\vec{a_2}-\vec{a_1})}{|\vec{b}|}|\end{array} \)

Shortest Distance between Skew Lines

A set of lines that do not intersect each other at any point and are not parallel are called skew lines (also known as agonic lines). Such a set of lines mostly exist in three or more dimensions.

For example, in the below diagram, RY and PS are skew lines among the given pairs.

Distance between Skew Lines

Distance Formula:

The distance between two lines of the form,

\(\begin{array}{l}\vec{l_{1}}= \vec{a_{1}}+t\vec{b_{1}}\end{array} \)

and

\(\begin{array}{l}\vec{l_{2}}= \vec{a_{2}}+t\vec{b_{2}}\end{array} \)

\(\begin{array}{l}d=\frac{(\vec{b_{1}}\times \vec{b_{2}}).(\vec{a_{2}}-\vec{a_{1}})}{\left | \vec{b_{1}} \times \vec{b_{2}}\right |}\end{array} \)

Solved Examples

Example 1: Find the distance between two parallel lines y = x + 6 and y = x – 2.

Solution: Given equations are of the form, y = mx + c

Here, m = 1, c₁ = 6, c₂ = -2

Formula: d = |c₁ – c₂|/√(1 + m²)

Therefore, d = 8/√2 or 5.65 Units.

Example 2: Find the shortest distance between lines

\(\begin{array}{l}\vec{r} = \hat{i} + 2\hat{j} + \hat{k} + \lambda ( 2\hat{i} + \hat{j} + 2\hat{k})\end{array} \)

and
\(\begin{array}{l}\vec{r} = 2\hat{i} – \hat{j} – \hat{k} + \mu ( 2\hat{i} + \hat{j} + 2\hat{k})\end{array} \)

Solution:

Using formula,

\(\begin{array}{l}d = |\frac{\vec{b} \times (\vec{a_2}-\vec{a_1})}{|\vec{b}|}|\end{array} \)

Here,

\(\begin{array}{l}|\vec{b} \times (\vec{a_2}-\vec{a_1})| = \begin{vmatrix} i & j & k\\ 2 & 1 &2 \\ 1 & -3 &-2 \end{vmatrix}\end{array} \)

= |4i + 6j – 7k|

= √101

And

\(\begin{array}{l}|\vec{b}|= 3\end{array} \)

Therefore, d = √101/3

Frequently Asked Questions

How to find the distance between two parallel lines?

Let two parallel lines be represented by y = mx+c₁ and y = mx+c₂. The distance between the lines is given by d = |(c₂-c₁)/√(1 + m²)|.

What do you mean by skew lines?

Lines that do not intersect each other at any point and are not parallel are called skew lines. These lines exist in three dimensions.

What is the shortest distance between two intersecting lines?

The shortest distance between two intersecting lines is equal to 0.