Distance from Point to Plane Calculator – Guide & Formulas
Calculate the perpendicular distance from a point to a plane in 3D space. Free online calculator with step-by-step solutions using the standard plane equation ax + by + cz + d = 0.
Calculate the perpendicular distance from any point to a plane in 3D space. Enter the plane coefficients and point coordinates to get instant results with step-by-step explanations.
Key Takeaway
Use the free Distance from Point to Plane Calculator to calculate the perpendicular distance from a point to a plane in 3d space. free online calculator with step-by-step solutions using the standard plane equation ax + by + cz + d = 0. Get instant results with step-by-step explanations.
How to Use the Distance from Point to Plane Calculator
- Enter the plane equation coefficients a, b, c, and d for ax + by + cz + d = 0.
- Enter the point coordinates (x₀, y₀, z₀).
- Click Calculate to find the perpendicular distance.
- Review the step-by-step solution showing the distance formula application.
The Formula
Variable Definitions
- a, b, c: Coefficients of the plane equation (normal vector components)
- d: Constant term in the plane equation
- x₀, y₀, z₀: Coordinates of the point
- D: Perpendicular distance from the point to the plane
Distance from Point (1, 2, 3) to Plane 2x + 3y + 6z - 12 = 0
Calculate the shortest distance from a point to a given plane.
- Step 1: Identify coefficients: a=2, b=3, c=6, d=-12. Point: (1, 2, 3).
- Step 2: Substitute into numerator: |2(1) + 3(2) + 6(3) - 12| = |2 + 6 + 18 - 12| = |14| = 14.
- Step 3: Calculate denominator: √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
- Step 4: Distance = 14 / 7 = 2 units.
Frequently Asked Questions
What is the distance from a point to a plane?
The distance from a point to a plane is the length of the shortest (perpendicular) line segment connecting the point to the plane. It is always measured along the normal direction to the plane.
How do I find the equation of a plane?
A plane can be defined by three non-collinear points, a point and a normal vector, or by intercepts. The standard form is ax + by + cz + d = 0, where (a,b,c) is the normal vector.
What happens when the distance is zero?
A distance of zero means the point lies exactly on the plane. The point satisfies the plane equation: ax₀ + by₀ + cz₀ + d = 0.
Can the distance be negative?
No, distance is always non-negative. The absolute value in the formula ensures the result is always zero or positive, regardless of which side of the plane the point is on.
What is the normal vector of a plane?
The normal vector is a vector perpendicular to the plane. For the plane ax + by + cz + d = 0, the normal vector is (a, b, c). It determines the orientation of the plane in 3D space.
How is this different from the distance between two points?
The distance between two points uses the Euclidean formula √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²). The point-to-plane distance uses the projection of the point onto the plane normal.
What if a, b, and c are all zero?
If a = b = c = 0, the equation does not represent a valid plane (it becomes d = 0, which is either always true or impossible). At least one of a, b, or c must be non-zero.
How do I convert from intercept form to standard form?
Intercept form: x/A + y/B + z/C = 1. Multiply through by ABC to get: BCx + ACy + ABCz = ABC, then rearrange to: BCx + ACy + ABCz - ABC = 0.
Can I use this formula in 2D?
Yes, in 2D the formula simplifies to D = |ax₀ + by₀ + c| / √(a² + b²) for the line ax + by + c = 0. It is the same principle applied to one fewer dimension.
What is the projection of a point onto a plane?
The projection is the closest point on the plane to the given point. It is found by moving from the point along the normal direction by the perpendicular distance.