Point of Intersection Calculator

Enter the coefficients and constants to get the coordinates of the intersection point, with steps & graph shown.

Add calculator to your Website

The point of intersection calculator calculates the point where two lines cross each other on an XY plane. Enter the equations and the calculator will calculate the intersection point coordinates in a 2D or 3D plane. Get to know whether the two lines are parallel or perpendicular.

Point of Intersection:

It is the point where two lines intersect each other.

Two non-parallel lines will always have a common point representing their intersection coordinates. And our intersection calculator will certainly figure out these points along with graphical interpretation.

Point of Intersection - Formula:

Finding points of intersection is a bit different when it comes to 2D and 3D planes. But the point of intersection calculator will readily calculate the coordinates no matter in which plane your lines are intersecting. If your goal comes up with manual calculations, follow the lead as under!

2D-Plane:

Standard Form:

If the equation is in the standard form as under:

\(A_{1}x + B_{1}y + C_{1} = 0\)

\(A_{2}x + B_{2}y + C_{2} = 0\)

Then you can find the point of intersection as:

\(x = \dfrac{B_{1}C_{2} - B_{2}C_{1}}{A_{1}B_{2} - A_{2}B_{1}}\)

\(y = \dfrac{C_{1}A_{2} - C_{2}A_{1}}{A_{1}B_{2} - A_{2}B_{1}}\)

Slope Intercept Form:

If you are having the point-slope form of the equation of lines as:

\(y = a_{1}x + b_{1}\) \(y = a_{2}x + b_{2}\)

Then our line intersection calculator will calculate intersection of two lines by using the formula below:

\(x = \dfrac{b_{2} - b_{1}}{a_{1} - a_{2}}\)

\(y = a_{1}\dfrac{b_{2} - b_{1}}{a_{1} - a_{2}} + b_{1}\)

3D Plane:

In three dimensional lane, we consider the parametric notations of the equations of two lines. Suppose two lines intersect in a 3D plane for which the equations are:

For Line 1:

\(x = x_{1}t + a_{1}\)

\(y = y_{1}t + b_{1}\)

\(z = z_{1}t + c_{1}\)

For Line 2:

\(x = x_{2}s + a_{2}\)

\(y = y_{2}s + b_{2}\)

\(z = z_{2}s + c_{2}\)

Now here after you calculate the value of either t or s. Just enter the determined value in the corresponding equation group and get the coordinates of the intersection point.

Angle of Intersecting Lines:

No doubt a couple of lines always make a certain measure of the angle by intersecting.

This can be calculated by using the following formula:

\(\tan\left(\theta\right) = \frac{m_{2} - m_{1}}{1 + m_{2} m_{1}}\)

How To Find Point of Intersection?

Let’s find the points of intersection of the two lines:

2x+3y = 4 —————–(1)

3x + 4y = 5 —————(2)

Now find the value of x from Eq (1)

2x = -3y + 4 —————(1)

Then

x = (-3y + 4)/2 ————-(3)

Put the value of x in Eq (2)

3(-3y + 4)/2 + 4y = 5

[(-9y + 12)/2]+ 4y = 5 y = 2

Put the value of x in Eq (2)

3x + 4(2) = 5

3x = 5 – 4(2)

3x = 5 – 8

x = -3/3

x = – 1

So the point of intersection is (-1, 2). For cross-verification, you may enter the equation coefficients in our points intersection of two lines calculator and check whether your results are accurate or not.

Working of Point of Intersection Calculator:

Using this find intersection of two lines calculator is quite straightforward! It requires you to enter the following inputs and get the coordinates of the coinciding point.

Inputs:

Select plane i.e; 2d or 3D
Enter the coefficient of x and y
Enter the constant values
Tap Calculate

Output:

Point of intersection
Azimuth
Graphical interpretation
Step-by-step calculations

Faqs:

Can the Line of Intersection Make a Right Angle?

Yes, the line of intersection can make a right angle to each other. It means the lines are perpendicular to each other.

What Is the Condition for the Intersection of Two Lines?

The necessary condition for the point of intersection of two lines is that this line lies on the same plane. Finding points of intersection, if the line does not lie on the same plan. These lines are not going to intersect each other.

What Is the Common Point of Intersection?

The intersecting lines share a common point and it is also called the point of interaction of two lines. You can find the points of intersection or common points of lines with the point of intersection calculator.