Equation of a Plane Calculator

Calculating the equation of a plane is a straightforward mathematical procedure that expands your knowledge of geometry from lines to planes in 3D space! Keep reading this article to learn:

• What is a plane (in geometry)?
• The equation of a plane.
• How to calculate the equation of a plane from a point and the normal vector.
• How to find the equation of a plane from 3 points.

And much more!

Planes, at least in geometry, don’t fly. A plane is a flat surface (with no curvature) that extends infinitely in two dimensions.

Planes are natural extensions of lines, obtained by adding a dimension. If a vector defines a straight line, then you can generate a plane by making infinite linear combinations of two vectors that aren’t parallel. And yes, you guessed it: adding another vector to the equation generates a three-dimensional space!

We can associate each plane with an equation. In its most generic form, the equation has three unknowns (matching the three coordinates $x$, $y$, and $z$ of Euclidean space).

where $a$, $b$, and $c$ are the coefficients, and $d$ is a constant for the plane.

The coefficients $a$, $b$, and $c$ can’t all be zero, or in other words:

To find the equation of a plane requires you to know two elements:

• A point on the plane ($P$); and
• The normal vector to the plane: a vector perpendicular to the surface ($\boldsymbol{n}$).

Three coordinates identify the point $P$:

$P$ anchors the plane in space, allowing it to “rotate” but fixing the distance of at least one of its points from the origin. At this point, we need to fix the orientation of the plane.

The normal vector is a fundamental clue that gives information on the plane’s orientation. However, on its own, it doesn’t tell us anything about the plane’s distance from the origin. The normal vector is:

Calculating the equation of a plane involves joining this information about the fixed point and the normal vector in relation to a point $P_0 = (x,y,z)$ that satisfies the equation of the plane (the point belongs to the plane).

What is the geometrical property that calculates the equation of a plane? The dot product: since both points lie on the plane, the vector connecting them lies on the plane as well. This vector is $P\!-\!P_0$. By definition, the normal vector is perpendicular to the plane; hence, their scalar product is zero. Let’s write this in the formula:

Let’s calculate the vector $P\!-\!P_0$:

Hence, the scalar product is:

If you have any doubts on this passage, visit our dot product calculator. We proceed with the calculations:

Let’s introduce a new quantity:

We conclude the calculations of the equation of a plane in 3D, then it becomes:

And then, isolating the constant term, we find the equation of a plane in its ultimate form:

If you don’t know the normal vector, you can still calculate the equation of a plane from 3 points on its surface. To do so, we need to add a step before calculating the equation of a plane.

Three points on the plane, let’s call them $P=(p_x,p_y,p_z)$, $Q=(q_x,q_y,q_z)$, and $R=(r_x,r_y,r_z)$, allow you to define two vectors, say $P-Q$ and $P-R$. These two vectors are not necessarily perpendicular; they can lie at any angle. However, if they aren’t parallel, their cross product is a vector perpendicular to the plane defined by the vectors. Let’s define the two vectors first:

And:

The result of the cross product $(P-Q)\times(P-R)$ is the normal vector $\boldsymbol{n}$. The result is pretty long:

Whelp! Anyway, we found our normal vector. Choose a point from the three above, and build the plane equation.

Why can we uniquely define the equation of a plane from three points? Imagine defining only two points on the plane: you’d identify a line on the plane, a line that is a rotation axis of the plane, around which the surface can rotate freely. Once you add a third point beyond the line, you “stop” this rotation.

It may happen that some of the parameters of the elements needed to define a plane are missing. Think of a normal vector in the x-y plane, thus with $c=0$. The generic equation for a plane in this situation would be:

That is:

This looks like the equation of a line. How come? Well, this plane is defined for all $z$ in a trivial way: the value of the z-coordinate doesn’t affect the x and y ones.

In the special case of two components of the normal vector that are zero, we’d find an equation in the following form, for example, for the vector $\boldsymbol{n}=\left\langle a,0,0 \right\rangle$:

Or $ax = d$. The resulting plane corresponds to a plane parallel to the yz plane at a distance $d/a$ from the origin.

Our equation of a plane calculator evaluates the equation of a plane in the blink of an eye as you insert parameters such as a point on the plane and the normal vector.

We understand that knowing the normal vector may not be that common: it’s more likely to know points on the plane. We added the possibility of inserting the three points needed to identify a plane: change option at the top of our tool!