# Line of Intersection of Two Planes Calculator

Created by Davide Borchia
Reviewed by Rijk de Wet
Last updated: Jan 18, 2024

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.

## What is a plane in geometry?

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:

$\footnotesize P_0=(p_x,p_y,p_z)$

And the normal vector is:

$\footnotesize \boldsymbol{n} = \left\langle a,b,c \right\rangle$

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

$\footnotesize d = ap_x +bp_y +cp_z ​$

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

$\footnotesize ax+by+cz=d$

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

🙋 We built a tool entirely dedicated to the equation of a plane: try our equation of a plane calculator!

## Before calculating the intersection of two planes: geometry of the problem

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!

🙋 You can learn about intersecting lines at our intersection of two lines calculator — and if your lines are parallel, perhaps you should visit our parallel lines calculator!

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. Respectively, these situations arise if the normal vectors match (hence the planes have the same orientation) and if $d_1=d_2$ (thus fixing the distance of the two planes) as well.

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.

## How to calculate the line of intersection of two planes?

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 that don't depend on the coefficients of the planes' equations.

The result is in the following form:

$\footnotesize \boldsymbol{r} = (k_1\boldsymbol{n}_1+k_2\boldsymbol{n}_2)+ \lambda(\boldsymbol{n}_1\times\boldsymbol{n}_2)$

where:

• $\boldsymbol{r} =\left\langle x,y,z \right\rangle$ — Vector containing a point on the line;
• $k_1$ and $k_2$ — Coefficients depending on the planes' equations;
• $\boldsymbol{n}_1$ and $\boldsymbol{n}_2$Normalized normal vectors (i.e. $|{\boldsymbol n}_1| = |{\boldsymbol n}_2| = 1$);
• $\boldsymbol{n}_1\times\boldsymbol{n}_2$ — Cross product of the normal vectors; and
• $\lambda$ — A parameter that can vary freely;

The coefficients $k_1$ and $k_2$ are:

$\footnotesize k_1 = \frac{d_1-d_2(\boldsymbol{n}_1\cdot\boldsymbol{n}_2)}{1-(\boldsymbol{n}_1\cdot\boldsymbol{n}_2)} \\[1em] k_2 = \frac{d_2-d_1(\boldsymbol{n}_1\cdot\boldsymbol{n}_2)}{1-(\boldsymbol{n}_1\cdot\boldsymbol{n}_2)}$

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

## How to find the intersection of two planes using the symmetric form of the line equation

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:

$\footnotesize x = 4y +\frac{7}{2} = 2z -\frac{1}{2}$

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

## How to find the line of intersection of two planes: an example

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

1. $-2x + 3y +4z = -1$; and
2. $2x -y -3z = 2$.

Let's find the two normal vectors:

1. For the first plane, we have $\boldsymbol{n}_1 = \left\langle -2,3,4\right\rangle$; and
2. For the second plane, we have $\boldsymbol{n}_2 = \left\langle 2,-1,-3\right\rangle$.

We normalize the vectors by dividing each component by the norm of the vector:

$\footnotesize \begin{split} |\boldsymbol{n}_1|&=\sqrt{(-2)^2+3^2+4^2}\\ &=\sqrt{4+9+16} = \sqrt{29} \end{split}$

We have:

$\footnotesize \boldsymbol{n}_1 = \left\langle-\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}} \right \rangle$

We find the norm for the second vector too:

$\footnotesize \begin{split} |\boldsymbol{n}_2|&=\sqrt{2^2+(-1)^2+(-3)^2}\\ &=\sqrt{4+1+9} = \sqrt{14} \end{split}$

And the normalized vector is:

$\footnotesize \boldsymbol{n}_2 = \left\langle \frac{2}{\sqrt{14}},-\frac{1}{\sqrt{14}},-\frac{3}{\sqrt{14}}\right \rangle$

We then calculate the coefficients we need to use in the equation for the line. For the first coefficient, we have the following:

$\footnotesize k_1 = \frac{-1-2(\boldsymbol{n}_1\cdot\boldsymbol{n}_2)}{1-(\boldsymbol{n}_1\cdot\boldsymbol{n}_2)}$

We compute the dot product of the two normal vectors — if you need a hand with this operation, visit our dot product calculator:

$\footnotesize \begin{split} \boldsymbol{n}_1\cdot\boldsymbol{n}_2 &= \left\langle-\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}} \right \rangle\\[1.2em] &\cdot \left\langle \frac{2}{\sqrt{14}},-\frac{1}{\sqrt{14}},-\frac{3}{\sqrt{14}}\right \rangle\\[1.2em] & = -\frac{4}{\sqrt{406}}-\frac{3}{\sqrt{406}}-\frac{12}{\sqrt{406}}\\[1.2em] & = -\frac{19}{\sqrt{406}}\simeq -0.943 \end{split}$

And find the value of the coefficient:

$\footnotesize k_1 = \frac{-1-2(-0.943)}{1+0.943}= 0.456$

And for the second coefficient, we have:

$\footnotesize k_2 = \frac{2+1(-0.943)}{1+0.943} = 0.544$

The last thing we need to find is the cross product between the normalized normal vectors. We will skip the steps here and give you only the final result — the cross product calculator will clear any doubt you can have.

$\footnotesize \begin{split} \boldsymbol{n}_1\!\times\!\boldsymbol{n}_2\!&=\!\left\langle -\frac{5}{\sqrt{406}},\frac{2}{\sqrt{406}},-\frac{4}{\sqrt{406}} \right\rangle \\[1em] &=\!\left\langle-0.2481,0.0993,-0.1985 \right\rangle \end{split}$

Let's compute the products of the coefficients by the respective normalized vectors:

$\footnotesize \begin{split}k_1\boldsymbol{n}_1 &= 0.456\left\langle-\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}} \right \rangle\\[1em] &=\left\langle-0.1693,0.254,0.3387\right \rangle \end{split}$

And:

$\footnotesize \begin{split} k_2\boldsymbol{n}_2 &= 0.544 \left\langle \frac{2}{\sqrt{14}},-\frac{1}{\sqrt{14}},-\frac{3}{\sqrt{14}}\right \rangle\\[1em] &=\left\langle 0.291,-0.1454,-0.4362 \right\rangle \end{split}$

We compute the sum of these two results:

$\footnotesize \begin{split} & k_1\boldsymbol{n}_1+k_2\boldsymbol{n}_2 \\ &= \left\langle-0.1693,0.254,0.3387\right \rangle\\ &\quad+ \left\langle 0.291,-0.1454,0.4362 \right\rangle\\[.5em] &=\left\langle 0.1215,0.1086,-0.0975\right\rangle \end{split}$

We can combine all these results, finally finding the line of intersection of the two planes in the parametric form:

$\footnotesize \begin{split} \boldsymbol{r}& = (k_1\boldsymbol{n}_1+k_2\boldsymbol{n}_2)+\lambda(\boldsymbol{n}_1\times\boldsymbol{n}_2)\\[.5em] &=\left\langle 0.1215,0.1086,-0.0975\right\rangle \\ &\quad+ \lambda\left\langle-0.2481,0.0993,-0.1985 \right\rangle \end{split}$

Ok, this expression is objectively clumsy; let's split it up so that we can see the individual equations cooperating for the definition of this line:

$\footnotesize \begin{split} x &= 0.1215-0.2481\lambda\\ y&=0.1086+0.0993\lambda\\ z&=-0.0975-0.1985\lambda \end{split}$

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

## How to use our line of intersection of two planes calculator

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 coefficients of any of the variables are not defined in your equations.

If you insert all the parameters of the planes' equation, we can also give you the equation of the line of the intersection of the two planes in symmetric form!

## FAQ

### Can the intersection of two planes be a point?

No. A point can't be the intersection of two planes: as planes are infinite surfaces in two dimensions if two of them intersect, the intersection "propagates" as a line. A straight line is also the only object that can result from the intersection of two planes. If two planes are parallel, no intersection can be found.

### How to find the line of intersection between two planes?

To find the line of intersection of two planes, you can follow these generic steps:

1. Isolate if possible, either z or y from the first plane's equation.
2. Substitute the expression in the second plane equation's corresponding variable.
3. Isolate x in the second plane's equation: you'll find an equation for the line in the xy or the xz plane.
4. If necessary, repeat the steps above for the other variable to find the equation in the other plane.

You are done: the generic resulting expression should be x = ay + b = mz + q.

### What is the line of intersection between the planes x + y = 0 and z = 3?

The line of intersection of the planes x + y = 0 and z = 3 is, in parametric form: r = ⟨0, 0, 3⟩ + λ⟨1/√2, -1/√2, 0⟩. To find this result:

1. Normalize the normal vectors:

n1 = ⟨1/√2, 1/√2, 0⟩ and n2 = ⟨0, 0, 1⟩.

2. Find the coefficients for the parametric equation: k1 = 0 and k2 = 3.

3. Compute the cross product of the vectors: n1 × n2 = ⟨1/√2, -1/√2, 0⟩.

4. Compute the following:

r = (k1n1 + k2n2) + λ(n1 × n2) = ⟨0, 0, 3⟩ + λ⟨1/√2, -1/√2, 0⟩.

### What is the parametric equation of the line of intersection of two planes?

The parametric equation of the line of intersection of two planes is an equation in the form r = (k1n1 + k2n2) + λ(n1 × n2).
where:

• n1 and n2Normalized normal vectors.
• k1 and k2 — Coefficients of the equation in the form ki = di - dj(n1 · n2)/(1 - (n1 · n2)) where d is the constant of the plane equation.
• λParameter of the equation.

The result is a vector equation that defines how the line evolves in each coordinate.

Davide Borchia
First plane
a
b
c
d
Second plane
a
b
c
d
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