Find Equation Of Plane With 3 Points Calculator

Easily calculate the equation of a plane (ax + by + cz = d) given the coordinates of three distinct points in 3D space with our find equation of plane with 3 points calculator.

Plane Equation Calculator

Point 1 (P1):

Point 2 (P2):

Point 3 (P3):

Point	X	Y	Z
P1	1	2	3
P2	0	1	0
P3	2	0	1

Input points used to define the plane.

What is a find equation of plane with 3 points calculator?

A find equation of plane with 3 points calculator is a tool used to determine the standard equation of a plane (in the form ax + by + cz = d) that passes through three distinct, non-collinear points in three-dimensional space. Given the coordinates (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) of these three points, the calculator computes the coefficients a, b, c, and the constant d.

This calculator is useful for students studying geometry, physics, or engineering, as well as professionals who need to define planes in 3D modeling, computer graphics, or spatial analysis. It automates the process of finding the plane equation, which involves vector cross products and algebraic manipulation.

Common misconceptions include thinking that any three points define a unique plane (they must be non-collinear) or that the order of points matters (it only affects the direction of the normal vector, which can be normalized).

Find Equation of Plane with 3 Points Calculator Formula and Mathematical Explanation

To find the equation of a plane passing through three points P1(x1, y1, z1), P2(x2, y2, z2), and P3(x3, y3, z3), we first define two vectors lying in the plane:

• Vector v1 = P2 – P1 = (x2-x1, y2-y1, z2-z1)
• Vector v2 = P3 – P1 = (x3-x1, y3-y1, z3-z1)

The normal vector n = (a, b, c) to the plane is perpendicular to both v1 and v2, and can be found by taking their cross product: n = v1 x v2.

The components of the normal vector (a, b, c) are calculated as:

• a = (y2-y1)(z3-z1) – (z2-z1)(y3-y1)
• b = (z2-z1)(x3-x1) – (x2-x1)(z3-z1)
• c = (x2-x1)(y3-y1) – (y2-y1)(x3-x1)

Once we have the normal vector (a, b, c), the equation of the plane can be written as a(x – x1) + b(y – y1) + c(z – z1) = 0. Expanding this, we get ax + by + cz – (ax1 + by1 + cz1) = 0. So, the equation is ax + by + cz = d, where d = ax1 + by1 + cz1.

If a, b, c, and d have a common factor, they are often divided by it to simplify the equation.

Variable	Meaning	Unit	Typical Range
(x1, y1, z1)	Coordinates of the first point	Length units	Any real number
(x2, y2, z2)	Coordinates of the second point	Length units	Any real number
(x3, y3, z3)	Coordinates of the third point	Length units	Any real number
a, b, c	Components of the normal vector to the plane	–	Any real number
d	Constant in the plane equation ax+by+cz=d	–	Any real number

Variables used in the find equation of plane with 3 points calculator.

Practical Examples (Real-World Use Cases)

Let's see how the find equation of plane with 3 points calculator works with examples.

Example 1:

Suppose we have three points: P1(1, 2, 3), P2(0, 1, 0), and P3(2, 0, 1).

• v1 = (0-1, 1-2, 0-3) = (-1, -1, -3)
• v2 = (2-1, 0-2, 1-3) = (1, -2, -2)
• a = (-1)(-2) – (-3)(-2) = 2 – 6 = -4
• b = (-3)(1) – (-1)(-2) = -3 – 2 = -5
• c = (-1)(-2) – (-1)(1) = 2 – (-1) = 3
• d = (-4)(1) + (-5)(2) + (3)(3) = -4 – 10 + 9 = -5

The equation of the plane is -4x – 5y + 3z = -5, or 4x + 5y – 3z = 5.

Example 2:

Consider points P1(2, -1, 1), P2(3, 2, -1), and P3(-1, 3, 2).

• v1 = (3-2, 2-(-1), -1-1) = (1, 3, -2)
• v2 = (-1-2, 3-(-1), 2-1) = (-3, 4, 1)
• a = (3)(1) – (-2)(4) = 3 – (-8) = 11
• b = (-2)(-3) – (1)(1) = 6 – 1 = 5
• c = (1)(4) – (3)(-3) = 4 – (-9) = 13
• d = (11)(2) + (5)(-1) + (13)(1) = 22 – 5 + 13 = 30

The equation of the plane is 11x + 5y + 13z = 30.

How to Use This Find Equation of Plane with 3 Points Calculator

1. Enter Coordinates: Input the x, y, and z coordinates for each of the three points (P1, P2, P3) into the respective fields.
2. View Results: The calculator will automatically compute and display the components of the normal vector (a, b, c), the constant d, and the final equation of the plane ax + by + cz = d as you enter the values.
3. Interpret Results: The "Primary Result" shows the simplified equation of the plane. The "Intermediate Results" show the values of a, b, c, and d.
4. Reset: Click the "Reset" button to clear the inputs and results and start with default values.
5. Copy: Click "Copy Results" to copy the equation and intermediate values to your clipboard.

The visualization shows the relative magnitudes of the normal vector components.

Key Factors That Affect Find Equation of Plane with 3 Points Calculator Results

• Coordinates of the Points: The most direct factor. Changing any coordinate of P1, P2, or P3 will change the orientation and position of the plane, thus altering a, b, c, and d.
• Collinearity of Points: If the three points are collinear (lie on the same line), they do not define a unique plane. In this case, the cross product v1 x v2 will be the zero vector (0, 0, 0), and our calculator will indicate that a unique plane cannot be determined (a=b=c=0).
• Order of Points for Vectors: The order in which you define vectors (e.g., P2-P1 vs P1-P2) will affect the direction of the normal vector (a, b, c will have opposite signs), but the final plane equation can be normalized to look the same (e.g., x+y+z=1 is the same plane as -x-y-z=-1).
• Numerical Precision: Very large or very small coordinate values might lead to precision issues in calculations, though our find equation of plane with 3 points calculator uses standard floating-point arithmetic.
• Distinct Points: The three points must be distinct. If two or more points are identical, you effectively have only two or one point(s), which is not enough to define a unique plane.
• Magnitude of Normal Vector: The calculated normal vector (a, b, c) can be any vector perpendicular to the plane. Its magnitude depends on the specific points chosen. The equation can be simplified by dividing a, b, c, and d by their greatest common divisor.

Frequently Asked Questions (FAQ)

What if the three points are collinear?

If the three points lie on the same straight line, they do not define a unique plane. Infinitely many planes can pass through a single line. In this case, the cross product of the vectors formed by the points will be zero (a=b=c=0), and the find equation of plane with 3 points calculator will indicate this.

Does the order of the points matter?

The order of points used to form the vectors v1 and v2 affects the direction of the normal vector (it might point "up" or "down" relative to the plane), but the resulting plane equation will represent the same plane, possibly with all coefficients having opposite signs.

What does ax + by + cz = d represent?

This is the standard form of the equation of a plane in 3D space, where (a, b, c) are the components of a vector normal (perpendicular) to the plane, and d is a constant related to the plane's distance from the origin (when the normal vector is normalized).

Can I use this calculator for points in 2D?

This calculator is specifically for 3D space (x, y, z coordinates). Three non-collinear points in 2D define a plane that *is* the 2D plane itself (like z=0 if they are in the xy-plane).

How is the normal vector calculated?

The normal vector is found by taking the cross product of two vectors that lie in the plane, formed by the three given points (e.g., P2-P1 and P3-P1). See our vector cross product calculator for more.

What if a, b, c, and d are all zero?

This would imply the three points are collinear or not distinct, and a unique plane is not defined by them.

Can the equation be simplified?

Yes, if a, b, c, and d share a common divisor, the equation is often simplified by dividing through by that divisor. Our find equation of plane with 3 points calculator attempts to provide a reasonably simplified form.

What if one of the normal vector components (a, b, or c) is zero?

If, for example, a=0, it means the normal vector is perpendicular to the x-axis, and the plane is parallel to the x-axis.

Related Tools and Internal Resources

• Vector Cross Product Calculator: Calculate the cross product of two vectors, used to find the normal vector.
• Distance Between Two Points Calculator: Find the distance between any two points in 3D space.
• Equation of a Line Calculator: Find the equation of a line given points or other properties.
• Matrix Determinant Calculator: The cross product can also be represented using a determinant.
• Understanding Vectors: Learn more about vectors and their properties.
• Plane Geometry Basics: An introduction to the geometry of planes.