Skew lines are a type of non intersecting lines in space. You can identify skew lines when you look at the floor of your room as shown below.

The edges formed by the walls with the floor form skew lines with the intersecting lines of the walls as shown in diagram. The skew lines are shown shaded with same colors.

Let us now look how the skew lines are different from parallel lines (the type of non intersecting lines you are already familiar with).

## Skew Lines Definition

Non coplanar lines in space that do not intersect are called skew lines. Segments and rays contained in skew lines are also skew. Let us identify the skew edges of the cube ABCDEFGH.

1. Edges AB and CD.                      2. Edges AB and GH.
3. Edges BC and AF.                       3. Edges BC and GH.
5. Edges CH and AF.                       6. Edges CH and BE.
7. Edges AH and BE.                       8. Edges AH and CD.
9. Edges DE and AF.                       10. Edges DE and GH.
11. Edges EF and GH.                    12. Edged EF and CD.
13. Edges FG and BE                     14. Edges FG and CD.
15. Edges GD and BE.                    16. Edges GD and AF.

You may also note any diagonal of a face is skew to the edges of the opposite face, like AC is skew to GD.
The angle between two lines the angle between two coplanar lines drawn parallel to the two lines.

## Skew Line Segments

Segments or rays of skew lines are also themselves skew. The skew edges shown in the above examples are indeed skew line segments of larger hidden skew lines.

## Skew Lines Examples

Let us consider the 3-D Plane.  Any line or line segment lying on yz - plane are all skew lines to x - axis.

Similarly lines lying totally on xy - plane are all skew to z axis and lines on xz - plane are all skew to the y axis.

2. Find all the segments skew to the edge CD in the triangular prism shown:

Segments AB and EF which lie in the face opposite CD are skew to edge CD.

## Skew Lines in Real Life

1. A road over a bridge crossing a railway track below.
2. The upright tree on one bank of a river, to the straight opposite Bank.

4. The Balcony railing is skew to the stick placed with its top touching the side wall.

You can find many such examples inside and outside your house or if you look around your class room.

## Distance between Skew Lines

The shortest distance between two skew lines is the length of the line segment perpendicular to both the lines.
It is possible to identify two parallel planes each containing one of the skew lines. In the example of cube, we identified AB and CD as two skew line segments. The plane ABEF containing the line AB and the plane CDGH containing the line CD are  parallel panes.

Hence the distance between the two skew lines AB and CD = distance between the parallel planes ABEF and CDGH.
= distance BC
Thus we can identify a parallel plane ax + by + cz + d = 0 containing one of the skew lines and a point P(x0, y0, z0) lying on the other skew line and use the formula to find the perpendicular distance of the plane from the point,
D = $\frac{|ax_{0}+by_{0}+cz_{0}+d|}{\sqrt{a^{2}+b^{2}+c^{2}}}$which would be the distance between the two skew lines.

### Solved Example

Question: Find the distance between the skew lines L1 and L2 whose parametric equations are given as follows:
L1 : x = 1 + 2t, y = 3t and z = 2 - t.
L2 : x = -1 + s, y = 4 + s, z = 1 + 3s.
Solution:

Step 1: Finding equation of a plane containing the line L2.
Let P1 and P2 be the two parallel planes containing L1 and L2. The common normal vector of these planes 'n' must be orthogonal to both v1 = <2, 3, -1>  (direction of L1)  and v2 = <1, 1, 3> (direction of L2).
The normal vector n = v1 x v2 = $\begin{vmatrix} i &j &k\\ 2& 3 &-1 \\ 1 &1 &3 \end{vmatrix}$ = i(9 + 1) -j(6 + 1) +k(2 - 3) = 10i -7j - k.
If we put s =0, in the equation of L2, we get a point on L2 as (-1, 4, 1).
Thus the equation of the plane P2 is,
a(x - x0) + B(y - y0) + c(z -z0) = 10
10(x + 1) -7(y - 4) -(z -1) = 0
10x - 7y - z + 39 = 0                         Equation of plane P2.

Step 2: Finding a point on line L1.
If we let t = 0 in the equation  of L1, we get the point on L1 as (1, 0, 2).

Step 3: Use distance formula to find the distance between the skew lines
Now we need to find the distance between the point (1, 0, 2) and the plane P2, using the formula

D = $\frac{|ax_{0}+by_{0}+cz_{0}+d|}{\sqrt{a^{2}+b^{2}+c^{2}}}$

= $\frac{|10(1)-7(0)-1(2)+39|}{\sqrt{10^{2}+(-7)^{2}+(-1)^{2}}}$

= $\frac{47}{\sqrt{150}}$ ≈ 3.84 units

The distance between the skew lines L1 and L2 ≈ 3.84 units.