Distance from Point to Plane Calculator

What is the Shortest Distance from a Point to a Plane?

The shortest distance from a point to a plane is the length of the perpendicular line segment from the point to the plane. This distance represents the minimum separation between the point and any point on the plane.

How to Calculate the Distance from a Point to a Plane

To find the shortest distance from a point to a plane, we follow these steps:

1. Identify the plane equation in the form \(Ax + By + Cz + D = 0\)
2. Determine the coordinates of the point \((x_0, y_0, z_0)\)
3. Apply the distance formula
4. Simplify and calculate the absolute value

Formula

The formula for the distance \(d\) from a point \((x_0, y_0, z_0)\) to a plane \(Ax + By + Cz + D = 0\) is:

\[ d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Where:

• \(A, B, C, D\) are coefficients of the plane equation
• \((x_0, y_0, z_0)\) are coordinates of the point
• \(|\cdot|\) denotes absolute value

Calculation Steps

1. Substitute the point coordinates and plane coefficients into the formula
2. Calculate the numerator: \(|Ax_0 + By_0 + Cz_0 + D|\)
3. Calculate the denominator: \(\sqrt{A^2 + B^2 + C^2}\)
4. Divide the numerator by the denominator

Example and Visual Representation

Let's find the distance from the point P(1, 2, 3) to the plane 2x - y + 2z - 4 = 0:

Step 1: Identify A = 2, B = -1, C = 2, D = -4, x₀ = 1, y₀ = 2, z₀ = 3

Step 2: \(|Ax_0 + By_0 + Cz_0 + D| = |2(1) + (-1)(2) + 2(3) + (-4)| = |2 - 2 + 6 - 4| = |2| = 2\)

Step 3: \(\sqrt{A^2 + B^2 + C^2} = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{9} = 3\)

Step 4: \(d = \frac{2}{3} \approx 0.667\)

In this diagram, the blue plane represents 2x - y + 2z - 4 = 0. The red point P(1,2,3) is shown above the plane. The red line segment represents the shortest distance from P to the plane, which we calculated to be approximately 0.667 units.

Special Cases

1. Distance to the xy-plane: For the xy-plane (z = 0), the formula simplifies to \(d = |z_0|\), where z₀ is the z-coordinate of the point.

2. Distance from the origin: For the origin (0, 0, 0), the formula becomes \(d = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}}\), where D is the constant term in the plane equation.