Distance Between Two Lines: Formula and Examples

Distance Between Two Lines: Formula and Examples

TL;DR

The distance between two lines is the length of the shortest segment joining them, measured perpendicular to both. For parallel lines ( ax + by + c_1 = 0 ) and ( ax + by + c_2 = 0 ), it is ( d = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} ); for skew lines in 3D, it uses the cross product. This article defines both cases and works through examples.

Why "The Distance Between Two Lines" Is Only One Number When The Lines Never Meet

The distance between two lines is the length of the shortest segment connecting them, and it is always measured perpendicular to both lines. Two lines fall into three cases: intersecting (distance 0), parallel (a constant perpendicular gap), and - in 3D only - skew (non-parallel lines that never meet, joined by a unique common perpendicular). This builds directly on the distance of a point from a line; for the plane it lives in, see coordinate geometry.

By the end you will identify the case, apply the right formula, and compute the distance for parallel lines in the plane and skew lines in space.

The Parallel-Lines Formula (2D)

Two lines are parallel when they have the same slope and never meet. Their perpendicular distance is the same everywhere, so you can measure it at any convenient point.

Write both lines with identical ( a ) and ( b ) coefficients:

[ ax + by + c_1 = 0 \quad \text{and} \quad ax + by + c_2 = 0 ]

Then the distance is:

[ d = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} ]

Here ( a ) and ( b ) are the shared coefficients of ( x ) and ( y ), and ( c_1, c_2 ) are the two constants. The absolute value keeps the distance positive regardless of order.

If the lines are given in slope form ( y = mx + c_1 ) and ( y = mx + c_2 ), rewrite as ( mx - y + c_1 = 0 ), giving:

[ d = \frac{|c_2 - c_1|}{\sqrt{m^2 + 1}} ]

The Skew-Lines Formula (3D)

In three dimensions, two non-parallel lines that never intersect are skew. Write the lines in vector form:

[ \vec{r_1} = \vec{a_1} + t \vec{b_1}, \quad \vec{r_2} = \vec{a_2} + s \vec{b_2} ]

Then:

[ d = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|} ]

The cross product ( \vec{b_1} \times \vec{b_2} ) is a vector perpendicular to both lines' directions; the dot product in the numerator projects the gap between the lines onto that common perpendicular.

Examples Of The Distance Between Two Lines

Example 1

Find the distance between ( 3x + 4y + 7 = 0 ) and ( 3x + 4y - 5 = 0 ).

The coefficients ( a = 3, b = 4 ) match, so the lines are parallel. Thus, ( c_1 = 7, c_2 = -5 ).

[ d = \frac{|c_2 - c_1|}{\sqrt{9 + 16}} = \frac{12}{5} = 2.4 \text{ units.} ]

Example 2

Find the distance between ( 2x + 4y = 5 ) and ( x + 2y = 1 ).

These lines do not share the same coefficients. Scale the second line first:

[ 2x + 4y = 2 \quad \text{and} \quad 2x + 4y - 5 = 0. ]

Now calculate:

[ d = \frac{|c_2 - c_1|}{\sqrt{4 + 16}} = \frac{3}{\sqrt{20}} \approx 0.671 \text{ units.} ]

Example 3

Find the distance between the lines ( y=2x+3 ) and ( y=2x−4 ).

Rewrite as ( 2x - y + 3 = 0 ) and ( 2x - y - 4 = 0 ):

[ d = \frac{|-4 - 3|}{\sqrt{4 + 1}} = \frac{7}{\sqrt{5}} \approx 3.13 \text{ units.} ]

Example 4

Two lines cross at the point (1,2). What is the distance between them?

The distance is zero. Intersecting lines share a point, so the shortest distance between them is 0.

Example 5

Find the distance between the skew lines ( \vec{r_1}=(0,0,0)+t(1,0,0) ) and ( \vec{r_2}=(0,1,1)+s(0,1,0) ).

Calculating gives:

[ d = 1 \text{ unit.} ]

Example 6

Are the lines with directions ( \vec{b_1}=(1,2,3) ) and ( \vec{b_2}=(2,4,6) ) skew?

The directions are parallel. Use the parallel-distance approach.

Conclusion