Line of Intersection of Two Planes Calculator

Calculating the line of intersection of two planes is not always as simple as computing the intersection of two lines. In this article, we will explain to you how to calculate it — with multiple approaches for all possible cases! Keep reading to learn:

• What is a plane, and what is the intersection of two planes?
• How to find the intersection of two planes: the line in parametric form.
• How to calculate the line of intersection of two planes in the symmetric form.
• Examples of calculation of the line of intersection between two planes.

A plane in geometry is a two-dimensional surface in a 3D space, a natural extension of the concept of line in 2D geometry. Planes are flat surfaces — their curvature is zero.

A plane is uniquely identified by a point and a normal vector. The point fixes the distance between the plane and the origin of the Cartesian space, while the normal vector (a vector perpendicular to the plane) fixes the orientation of the plane in space.

Let's meet these two elements. The point on the plane is:

And the normal vector is:

To describe the plane in one equation, we compute a constant for the plane, $d$:

and then we write the equation of the plane in the Cartesian space with this neat formula:

Above, $x$, $y$, and $z$ are free to change, but only the combinations that are points on the plane satisfy the equation.

Two planes can intersect each other (unless, of course, they are parallel). When two planes intersect, a line in space is the result. This is a logical extension of the intersection of two lines in a 2D space, a much easier concept to understand:

When two lines (single-dimensional objects) intersect, we find a point, an object with zero dimensions. We can meet this reduction in dimensionality when considering planes, too. A plane is a two-dimensional object, so we expect the intersection of two planes to be a single-dimensional object, which is a line. This is indeed what we find!

As we said before, when two planes are parallel, we can't find an intersection, and they are either separated by a given distance or totally coincide. These situations arise when their normal vectors match (hence, the planes have the same orientation).

Finding the line of intersection of two planes involves dealing with lines in the 3D space: this is not a straightforward task! We can find the line of intersection of two planes in two ways; let's meet the first one.

In this section, we will calculate the line of intersection of two planes as a parametric equation. This method follows a simple sequence of steps:

1. Calculate the cross product of the normal vectors of both planes: $\boldsymbol{r} = \boldsymbol{n}_1 \times \boldsymbol{n}_2$. It is the directional vector of the line.

2. Find a common point for both planes, $P_0 = (x_0, y_0, z_0)$.

3. The equation of the resulting line of intersection is the following: $l: (x_0, y_0, z_0) + \lambda\boldsymbol{r}$. Here, $\lambda$ is a free parameter that can be any real number.

4. Assuming that the coordinates of the directional vector are $r_x$, $r_y$, and $r_z$, we can also rewrite the equation of the line into the separate components:

   • $x(\lambda) = x_0 + \lambda r_x$
   • $y(\lambda) = y_0 + \lambda r_y$
   • $z(\lambda) = z_0 + \lambda r_z$

By changing the value of parameter $\lambda$, we can generate every possible point on the line and thereby fully describe the line of intersection of our two planes.

If you are wondering how to find the common point $P_0$, we will describe the typical procedure in the following section with a computational example.

We can also find the intersection of two planes in symmetric form. This result may be more familiar than the parametric form. In the symmetric form, we equate $x$ to the other two spatial coordinates without needing an additional parameter like $\lambda$.

To do so, we perform a series of substitutions and rearrangements of the planes' equations. In this case, there is no one-size-fits-all list of instructions. We'd better illustrate this process with an example.

Take these two planes:

• $2x - 4z = -1$; and
• $x-2y+z = 2$.

As you can see, in the first equation, there is no coefficient for $y$: this means that the coordinates of points on this plane don't depend on $y$. To find the equation of the line, follow these steps:

1. Isolate the variable $z$ in the first equation: $z = \frac{1}{2}x + \frac{1}{4}$.

2. Substitute the equation found in the previous step to $z$ in the second equation: $x - 2y + (\frac{1}{2}x + \frac{1}{4}) = 2$.

3. Compute the sums: $(1-\frac{1}{2})x -2y = 2 - \frac{1}{4}$, i.e., $\frac{1}{2} x - 2y = \frac{7}{4}$.

4. Isolate $x$ in the equation above to obtain $x = 4y +\frac{7}{2}$: this is how we find the line of intersection of the two planes — or even better, its projection in the xy plane.

5. To find the projection of the intersection line in the xz plane, we can simply use the first plane equation: $x = 2z -\frac{1}{2}$.

The calculated intersection of the two planes in symmetric form is then:

Only the $x, y, z$ combinations that define points on the line will simultaneously satisfy these two equations!

Let's try the procedure to find the parametric equation of a line with a practical example. Take these planes:

• $A\!: -2x + 3y + 4z = -1$; and

• $B\!: 2x - y - 3z = 2$.

Let's find the two normal vectors:

• For the first plane, we have $\boldsymbol{n}_1 = \left\langle -2,3,4\right\rangle$; and

• For the second plane, we have $\boldsymbol{n}_2 = \left\langle 2,-1,-3\right\rangle$.

In the next step, we evaluate the cross-product of these normal vectors. It is the directional vector of the line of intersection:

Next, let's find the common point of both planes. We can start by setting one of the coordinates to a fixed value, e.g., $x_0 = 0$. In general, you can choose any of the coordinates.

By doing so, we can rewrite the equations of planes as a system of equations without the $x$ component:

We can find the solution to this set by using our system of equations calculator. The result is a pair: $y=1$ and $z=-1$.

This means that our common point of both planes is $P_0 = (0, 1, -1)$.

Finally, we can write the equations of the line of intersection in the parametric form:

Or split it into individual equations for all coordinates:

We vary $\lambda$ and find combinations of points on the line of intersection.

Our line of intersection of two planes calculator allows you to find the line of intersection in parametric form for every possible combination of non-parallel planes. Simply insert the parameters, using $0$, if the coefficient of any variable is not defined in your equations.

And that's all! The equation of the line of intersection will appear at the bottom of the calculator. In the case of parallel planes, you can check if they are separate or coincide.