The calculator measures the shortest distance between a point in three-dimensional space and a geometric plane. This tool is vital for anyone involved in designing or analyzing objects in 3D environments, offering a quick way to determine distances without manual calculations.
Formula of Shortest Distance From Point to Plane Calculator
To find the shortest distance from a point to a plane, we use the following formula:
- Given:
- A point in 3D space with coordinates (x0, y0, z0)
- The equation of the plane in the standard form Ax + By + Cz + D = 0
- Formula: d = (|Ax0 + By0 + Cz0 + D|) / sqrt(A^2 + B^2 + C^2)
- d represents the distance from the point to the plane.
- (A, B, C) are the coefficients of the plane equation, indicating the direction of the normal vector to the plane.
- (x0, y0, z0) are the coordinates of the point.
- D is the constant term in the plane equation.
- sqrt(A^2 + B^2 + C^2) is the magnitude of the normal vector to the plane.
Table of General Terms
Here is a table of general terms to help understand and apply the formula:
| Term | Definition | Relevance |
|---|---|---|
| Point | A specific location in 3D space defined by coordinates (x, y, z). | Essential for determining the starting point of measurement. |
| Plane | A flat, two-dimensional surface extending infinitely in 3D space. | The target from which the distance is measured. |
| Normal Vector | A vector perpendicular to the plane. | Important for understanding plane orientation and calculating distance. |
Example of Shortest Distance From Point to Plane Calculator
Consider a point P with coordinates (2, 3, 5) and a plane given by the equation x + 2y – 3z + 6 = 0. We will calculate the shortest distance from point P to this plane using the formula provided:
Given:
- Point P(2, 3, 5)
- Plane equation: x + 2y – 3z + 6 = 0
Using the formula for calculating the distance: d = (|Ax0 + By0 + Cz0 + D|) / sqrt(A^2 + B^2 + C^2)
First, substitute the values into the equation: A = 1, B = 2, C = -3, D = 6, and the coordinates of point P are x0 = 2, y0 = 3, z0 = 5.
Calculate Ax0 + By0 + Cz0 + D: = 12 + 23 + (-3)*5 + 6 = 2 + 6 – 15 + 6 = -1
Take the absolute value of the result: |Ax0 + By0 + Cz0 + D| = |-1| = 1
Next, calculate the denominator sqrt(A^2 + B^2 + C^2): = sqrt(1^2 + 2^2 + (-3)^2) = sqrt(1 + 4 + 9) = sqrt(14)
Finally, substitute back into the formula to find the distance d: d = 1 / sqrt(14) = 0.267
Thus, the shortest distance from point P to the plane is approximately 0.267 units.
Most Common FAQs
Yes, this calculator is versatile and can be used with any point and plane in three-dimensional space.
The calculator is highly accurate, using exact mathematical principles to ensure precise results.