Equation of a Plane Calculator – Guide & Formulas
Find the equation of a plane in 3D space. Free online calculator from three points, a point and normal vector, or intercept form with step-by-step solutions.
Find the equation of a plane in 3D space using three points, a point and normal vector, or intercepts. Enter your data to instantly get the plane equation with step-by-step solutions.
Key Takeaway
Use the free Equation of a Plane Calculator to find the equation of a plane in 3d space. free online calculator from three points, a point and normal vector, or intercept form with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Equation of a Plane Calculator
- Select input mode: Three Points, Point + Normal, or Intercepts.
- Enter the required coordinates or values.
- Click Calculate to find the plane equation in standard and general form.
- Review the normal vector, distance from origin, and step-by-step derivation.
The Formula
Variable Definitions
- a, b, c: Components of the normal vector to the plane
- d: Constant term (related to distance from origin)
- (x, y, z): Any point on the plane
- n⃗: The normal vector perpendicular to the plane surface
Plane Through Three Points: (1,0,0), (0,1,0), (0,0,1)
Find the equation of a plane passing through three non-collinear points.
- Step 1: Form two vectors: v₁ = P₂ − P₁ = (−1, 1, 0) and v₂ = P₃ − P₁ = (−1, 0, 1).
- Step 2: Compute normal: n = v₁ × v₂ = (1·1 − 0·0, 0·(−1) − (−1)·1, (−1)·0 − 1·(−1)) = (1, 1, 1).
- Step 3: Plane equation: 1(x − 1) + 1(y − 0) + 1(z − 0) = 0 → x + y + z − 1 = 0.
- Step 4: Final form: x + y + z = 1, or equivalently x/1 + y/1 + z/1 = 1 (intercept form).
Frequently Asked Questions
What is the equation of a plane?
The equation of a plane is ax + by + cz + d = 0, where (a,b,c) defines the normal direction and d determines the offset from the origin. Every point (x,y,z) satisfying this equation lies on the plane.
How do three points define a plane?
Three non-collinear points uniquely determine a plane. Two vectors formed from the points are crossed to produce the normal vector, which then defines the plane orientation.
What is a normal vector?
A normal vector is perpendicular to the plane. Any vector lying in the plane is orthogonal to the normal. The normal determines the plane's orientation in 3D space.
When do three points NOT define a plane?
If the three points are collinear (lie on the same line), they do not uniquely define a plane — infinitely many planes can pass through a single line.
What is the intercept form?
Intercept form: x/A + y/B + z/C = 1, where A, B, and C are the x, y, and z intercepts respectively. It is useful when you know where the plane crosses each axis.
How do I find the distance from the origin to a plane?
Distance = |d| / √(a² + b² + c²). This is the shortest distance from the origin to the plane along the normal direction.
Can two planes be parallel?
Yes. Two planes are parallel if their normal vectors are scalar multiples of each other. They never intersect and maintain a constant distance apart.
How do I find the angle between two planes?
The angle between two planes equals the angle between their normal vectors: θ = arccos(|n₁ · n₂| / (|n₁| × |n₂|)).
What is the cross product used for?
The cross product of two vectors in the plane produces the normal vector. This is essential when deriving the plane equation from three points.
Can a plane be vertical?
Yes. A vertical plane has a normal vector with zero z-component (a, b, 0). The plane equation becomes ax + by + d = 0, extending infinitely in the z-direction.