Intersection Of Two Lines

At most, only one point of intersection can be found between two separate lines. In order to locate the point where two lines meet, we require the general form of the two equations, which is represented by the notation a1x + b1y + c1 = 0 and a2x + b2y + c2= 0 respectively. If the lines are not parallel to one another, then only then will they intersect. Examples of real-world objects that feature intersecting lines include, but are not limited to, a pair of scissors, a folding chair, a road cross, a signboard, and many others. In this abridged version of a larger course, we will focus on learning the specifics of locating the place where two lines intersect.

Meaning of  intersection of two lines 

The term “intersecting lines” refers to the relationship between two lines that has exactly one point in common. One point is shared by both of the lines that intersect. The point at which all of the lines that intersect with one another have a common point is referred to as the point of intersection. There will be a point of intersection between the two co-planar straight lines that are not parallel to one another. The point O, which serves as the intersection of lines A and B, is referred to as the point of intersection.

Finding the point where two lines intersect

Let’s have a look at the following scenario, we have been provided with two lines, L1 and L2, and we are instructed to locate the spot where these lines intersect. Solving two simultaneous linear equations is required in order to determine the value of the point of intersection.

Let us write out in general form the equations that describe the two lines as follows: 

a1x + b1y + c1 = 0 

a2x + b2y + c2= 0 

Let’s say that the point of intersection is ( x0, y0) for the moment. Thus,

a1x0 + b1y0 + c1 = 0 

a2x0 + b2y0+ c2 = 0 

The following results can be obtained by applying Cramer’s rule to this system:

x0/(b1c2-b2c1) = -y0/(a1c2-a2c1)= 1/(a1b2-a2b1)

We can calculate the point of intersection (x0, y0) using this relation, which tells us that it is as follows:

(x0,y0) ={(b1c2-b2c1)/(a1b2-a2b1),(c1a2-c2a1)/(a1b2-a2b1)}

The angle of intersection 

Consider the figure that follows in order to calculate the angle formed by the intersection of two lines:

The slope-intercept form of the equations for the two lines is as follows:

y= -(a1/b1)x +(c1/b1) =m1x+c1

y= -(a1/b1)x +(c2/b2) =m2x+c2

Take note in the preceding picture that equals θ=θ2-θ1, and that as a result,

tanθ=tan(θ2-θ1)=

(tanθ2-tanθ1)/(1+tanθ2tanθ1)

= (m2-m1)/(1+m2m1)

Conventionally, we would only be concerned with the acute angle between the two lines; as a result, we require tan to be a quantity that has a positive value.

Therefore, in the expression that was just discussed, if the expression (m2-m1)/(1+m2m1), ends up having a negative value, this would be the tangent of the obtuse angle that exists between the two lines. Therefore, in order to calculate the acute angle that exists between the two lines, we use the magnitude of this expression.

Because of this, the acute angle that exists between the two lines is.

θ=tan-1|(m2-m1)/(1+m2m1)| 

Because of this relation, it is straightforward to derive the criteria that must be met by m1 and m2 for the two lines L1 and L2 to be either parallel or perpendicular to one another.

The factors that determine whether two lines are parallel or perpendicular to one another

It should come as no surprise that if the lines are parallel, then θ= 0 and m1 = m2, as the slope of parallel lines must be the same.

It is necessary that, θ =π/2 so that cotθ= 0 for the two lines to be considered perpendicular to one another. This can take place if 1 + m1 m2= 0 or if m1m2 = 1. 

The slope of this line can be calculated using the formula m = -a/b if the lines L1 and L2 have the general form ax+ by + c = 0.

•The prerequisite for establishing parallelism between two lines

Therefore, the following must hold true for L1 and L2 in order for them to be parallel:

m1=m2=⇒-a1/b1 = -a1/b2 ⇒a1/b1 = a2/b2

•The prerequisite for the perpendicularity of two lines.

The following must hold true for L1 and L2 to be in a perpendicular relationship:

m1m2=-1 ⇒(-a1/b1)(-a1/b2) =-1

                ⇒a1a2+b1b2=0

Conclusion

At most, only one point of intersection can be found between two separate lines. In order to locate the point where two lines meet, we require the general form of the two equations, which is represented by the notation a1x + b1y + c1= 0 and a2x + b2y + c2= 0 respectively. If the lines are not parallel to one another, then only then will they intersect. The term “intersecting lines” refers to the relationship between two lines that has exactly one point in common. One point is shared by both of the lines that intersect. The point at which all of the lines that intersect with one another have a common point is referred to as the point of intersection. It should come as no surprise that if the lines are parallel, then θ= 0 and m1 = m2, as the slope of parallel lines must be the same.