**This distance from point to plane calculator computes the shortest distance from a point (x₁, y₁, z₁) to a plane the equation Ax + By + Cz + D = 0, using formula: Distance = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²).**

**Consider a plane defined by the equation:**

**3x + 2y - z + 4 = 0**

And a point P(2, -1, 3) in space.

## Distance from Point to Plane Calculator

Plane Equation | Point Coordinates | Distance (units) |
---|---|---|

x + y + z = 1 | P(0, 0, 0) | 0.577 |

2x – y + z = 4 | P(1, 1, 1) | 1.633 |

3x + 2y – z = 6 | P(2, -1, 3) | 2.408 |

x – 2y + 2z = 3 | P(0, 2, 1) | 1.225 |

4x + 4y + 4z = 12 | P(1, 1, 1) | 0.000 |

For example,

## Distance from Point to Plane Formula

The **formula** for calculating the distance from a point to a plane is:

**|Ax₀ + By₀ + Cz₀ + D|
d = ───────────────────────
√(A² + B² + C²)**

Where:

- (x₀, y₀, z₀) are the coordinates of the point
- Ax + By + Cz + D = 0 is the equation of the plane
- A, B, and C are the
**coefficients**of the plane equation - D is the
**constant term** - |…| represents the
**absolute value** - √ represents the
**square root**

Given a plane **2x – 3y + 4z + 6 = 0** and point **P(1, 2, -1)**:

**d = |2(1) - 3(2) + 4(-1) + 6|
─────────────────────────
√(2² + (-3)² + 4²)
d = |2 - 6 - 4 + 6|
───────────────
√(4 + 9 + 16)
d = |-2|
─────
√29
d = 2/√29 ≈ 0.371 units**

## How to find Distance from Point to Plane?

Following these detailed steps:

**Identify**the plane equation in standard form (Ax + By + Cz + D = 0)**Note**the point coordinates (x₀, y₀, z₀)**Extract**the coefficients A, B, C, and D**Substitute**values into the formula**Calculate**the numerator and denominator separately**Perform**the final division

Given plane: **x + 2y – 2z – 4 = 0** and point **P(3, 0, 1)**:

- Identify coefficients:
- A = 1
- B = 2
- C = -2
- D = -4

- Calculate numerator:
**|1(3) + 2(0) – 2(1) – 4| |3 + 0 – 2 – 4| |-3| = 3** - Calculate denominator:
**√(1² + 2² + (-2)²) √(1 + 4 + 4) √9 = 3** - Final calculation:
**d = 3/3 = 1 unit**

## Sources and References

- MIT OpenCourseWare – Linear Algebra: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
- Paul’s Online Math Notes – Distance to Planes: https://tutorial.math.lamar.edu/

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