Two lines are said to be intersecting lines if they have a common point. It is to be noted that if two straight lines intersect each other, they can intersect at exactly point and that point is said to be the intersecting point of those two lines.

In the figure, there are two lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are two straight lines and P is a common point. Here, we say that the straight lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are intersecting at the point P and the point P is called the intersecting point of those two lines.

In a two dimensional plane, on the basis of this concept, any pair of straight lines can be categorized into the following cases:
1) Intersecting Lines
2) Non-intersecting Lines
3) Coincident Lines

It is to be noted that two or more lines can intersect at a single point. In such situations, we say that those lines are concurrent. It is to be noted that any pair of concurrent lines are intersecting lines, but pairs of intersecting lines may not concurrent. This fact is illustrated in the following diagram.

## Intersection of Two Lines

It is clear that the intersection of two straight lines is always a point and two lines can intersect at only one point, or they may coincident with each other.
The general equation of a straight line is of the form ax + by + c = 0,
where a, b, and c are real numbers. Now, consider two straight lines

a$_1$x + b$_1$y + c$_1$ = 0
a$_2$x + b$_2$y + c$_2$ = 0

Let P(x$_o$, y$_o$) be the intersecting point. Then, as the point is on both the lines, we have
a$_1$x$_o$ + b$_1$y$_o$ + c$_1$ = 0 (1)
a$_2$x$_o$ + b$_2$y$_o$ + c$_2$ = 0 (2)

## Find Intersection of Two Lines

Similarly,

From the above discussion, we have

This can also be written as

The calculations can be easily remembered using the following pattern:

For any two straight lines
a$_1$x + b$_1$y + c$_1$ = 0
a$_2$x + b$_2$y + c$_2$ = 0

we can consider the following three cases:

Case 1: The two lines intersects each other when

a$_1$b$_2$ - a$_2$b$_1$ $\neq$ 0

$\rightarrow$ a$_1$b$_2$ $\neq$ a$_2$b$_1$

$\rightarrow$ $\frac{a_1}{a_2}$ $\neq$ $\frac{b_1}{b_2}$

Case 2: The two lines coincident with each other when

b$_1$c$_2$ - b$_2$c$_1$ = 0, c$_1$a$_2$ - c$_2$a$_1$ = 0, and a$_1$b$_2$ - a$_2$b$_1$ = 0

$\rightarrow$ b$_1$c$_2$ = b$_2$c$_1$, c$_1$a$_2$ = c$_2$a$_1$, and a$_1$b$_2$ = a$_2$b$_1$

$\rightarrow$ $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$, $\frac{c_1}{c_2}$ = $\frac{a_1}{a_2}$, and $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$

$\rightarrow$ $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$

Case 3: The two given lines do not intersect when

b$_1$c$_2$ - b$_2$c$_1$ $\neq$ 0, c$_1$a$_2$ - c$_2$a$_1$ $\neq$ 0, and a$_1$b$_2$ - a$_2$b$_1$ $\neq$ 0

$\rightarrow$ b$_1$c$_2$ $\neq$ b$_2$c$_1$, c$_1$a$_2$ $\neq$ c$_2$a$_1$, and a$_1$b$_2$ = a$_2$b$_1$

$\rightarrow$ $\frac{b_1}{b_2}$ $\neq$ $\frac{c_1}{c_2}$, $\frac{c_1}{c_2}$ $\neq$ $\frac{a_1}{a_2}$, and $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$

$\rightarrow$ $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ $\neq$ $\frac{c_1}{c_2}$

In this case, the two lines never intersect.

## Intersecting Lines are Skew

Two lines in a 3-dimensional space are said to be skew if the two lines do not intersect and are not parallel.
Equivalently saying, two lines can be skew only if they are not coplanar as the lines on a plane either intersect each other or parallel.
The following figure shows two skew lines in a space.

Consider the Lines

L$_1$:x = -1 + t, y = 3 + 4t, z = 6 + 5t, t $\epsilon$ R

L$_2$:x = 4 - s, y = 17 + 2s, z = 30 - 5s, t $\epsilon$ R

The two lines have direction rations (1, 4, 5) and (-1, 2, -5). Clearly, the two lines are not parallel.
 L$_1$ L$_2$ x = -1 + t x = 4 - s y = 3 + 4t y = 17 + 2s z = 6 + 5t z = 30 - 5s

Now, when we equate x  and y coordinates, we get

-1 + t = 4 - s and 3 + 4t = 17 + 2s

$\rightarrow$ t + s = 5 and 4t - 2s = 14

When we solve, we get

Now, the z coordinates are not equal. For

z = 6 + 5t = 6 + 5(4) = 26

z = 30 - 5s = 30 - 5(1) = 25

This shows that the lines do not intersect.  So, they are skew lines.

## Intersecting Line Segments

We can say two line segments intersect if the lines containing them intersect on a common point of those two lines. Thus, it leads to three cases for line segments in two dimensional plane, as explained below:
1)    The two line segments intersect at a common point
2)    The lines containing the line segments intersects at a point external to the line segments
3)    The lines containing the line segments do not intersects

In case of three dimensional space, we can decide whether two line segments intersect or not using the value of the parameter for the intersecting points. Consider two line segments $\overline{AB}$  and $\overline{CD}$ with end points A(x$_1$, y$_1$, z$_1$), B(x$_2$, y$_2$, z$_2$), C(x$_3$, y$_3$, z$_3$) and D(x$_4$, y$_4$, z$_4$). Then, the lines containing the two line segments can be written in vector form, respectively, as
L$_1$ : (x, y, z) = (x$_1$, y$_1$, z$_1$) + t(x$_2$ - x$_1$, y$_2$ - y$_1$, z$_2$ - z$_1$), t $\epsilon$ R
L$_2$ : (x, y, z) = (x$_3$, y$_3$, z$_3$) + s(x$_4$ - x$_3$, y$_4$ - y$_3$, z$_4$ - z$_3$), t $\epsilon$ R
There are four cases to consider for the intersection of two lines in R$^3$.

Intersecting Lines
Case 1: The lines are not parallel and intersect at a single point.
Case 2: The lines are coincident, meaning that the two lines are identical. In this case, there are an infinite number of points of intersection. In other words, every point on the line is an intersecting point.

Non-intersecting Lines
Case 3: Two lines are parallel and there is no intersection.
Case 4: The two lines are not parallel, but there is no intersection. Such lines are called skew lines.
If  P(x, y, Z) is the intersecting point of the two line segments, it will be on both the line segments. Thus, for that point, we have
0 $\leq$ t, s $\leq$ 1

## Perpendicular Lines and Intersecting Lines

In case of a two dimensional plane, two intersecting lines are said to be perpendicular lines if they intersect at right angles.

In the figure, the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are perpendicular to each other intersecting at the point P. We can write
$\overleftrightarrow{AB}$ $\perp$ $\overleftrightarrow{CD}$
In three dimensional space, if two lines are perpendicular, the dot product of their direction rations is zero. It is to be noted that the perpendicular lines need not intersect.

For example, consider the lines
L$_1$ : (x, y, z) = (2t, -1, 1), t $\epsilon$ R

L$_2$ : (x, y, z) = (-1, 2s, 2), s $\epsilon$ R

Line L$_1$ is the intersection of the planes y = -1 and z = 1;  and the line L$_2$ is the intersection of the planes x = -1 and z = 2. The direction ratios of L$_1$ are (2, 0, 0)  and the direction ratios of L$_2$ are (0, 2, 0). Since

(2, 0, 0) x (0, 2, 0) = 2(0) + 0(2) + 0(0) = 0

The two lines are perpendicular. But, it should be noted that the two lines never intersect as the z-coordinates of the points on both the lines never equal. Clearly, these two lines are skew lines.

## Non Intersecting Lines

In a two dimensional plane, if two lines are non-intersecting lines, they should be parallel lines having no common point. But, in a three dimensional space, if two lines are non-intersecting, they come under the following two cases.
1) The two lines can be parallel
2) The two lines can be skew lines

## Intersection of a Line and a Plane

Let L be a line and $\pi$ be a plane in a three dimensional space.

The line L and the plane $\pi$ may intersect in three different ways:

Case 1: The line L intersects the plane $\pi$ at exactly one point P.
Case 2: The line L does not intersect the plane $\pi$ and is parallel to the plane $\pi$.  In this case, there is no point of intersection.
Case 3: The line L lies on the plane $\pi$.

Example:
Let the line L be

(x, y, z) = (-1 + 2t, -1 + t, 1 + 3t), t $\epsilon$ R

and the plane be

$\pi$ : z = 1.

Since z = 1, at the intersection point, we have

1 + 3t = 1

$\rightarrow$ t = 0

Thus, the intersecting point is

(-1 + 2t, -1 + t, 1 + 3t)

=(-1 + 3(0), -1 + 0, 1 + 3(0))

= (-1, -1, 1)

## Example of Intersecting Lines

Consider the two lines
2x + 3y = 5 (1)
x + 5y = 6 (2)
These two lines intersect at a single point as
$\frac{2}{1}$ $\neq$ $\frac{3}{5}$ Let us find out the intersecting points. For,

(2x + 3y) - 2(x + 5y) = 5 - 2(6)
$\rightarrow$ 2x + 3y - 2x - 10y = 5 -12
$\rightarrow$ -7y = -7
$\rightarrow$ y = 1

From (2)

x + 5y = 6
$\rightarrow$ x + 5(1) = 6
$\rightarrow$ x = 1

Thus, the intersecting point is (1, 1).