Understanding the Shortest Distance from a Point to a Plane

The concept of shortest distance from a point to a plane in three dimensional space is frequently used in mathematics and geometry.
This shortest distance is the perpendicular distance from the point to the plane. It is also referred to as the normal distance.

Defining a Point and a Plane

A point in three-dimensional space is represented by its Cartesian coordinates as (x, y, z).
A plane is defined by its standard equation form; ax + by + cz = d.

If P(x1,y1,z1) is a point in space and the equation of the plane is ax+by+cz=d, the shortest distance D from the point to the plane is given by the formula:

D =

ax1 + by1 + cz1 - d

/ √(a^2 + b^2 + c^2)

The numerator ax1 + by1 + cz1 - d gives the magnitude of the normal vector formed between the point and the plane.
The denominator √(a^2 + b^2 + c^2) calculates the length of the normal to the plane.

The normal to a plane is a vector that is perpendicular to it.
The shortest or perpendicular distance from a point to a plane can only be measured along the normal of the plane.

Substitute the coefficients (a, b and c) of the plane equation and the coordinates (x1, y1 and z1) of the point into the shortest distance formula.
This will give you the shortest distance from the given point to the specified plane.

Just like mastering transformations, practice is also key to improving shortest distance calculations.
Start by practicing problems with simple plane equations and points, and gradually work your way up to more complex ones.
Sketching the point and plane can aid in visualising this three-dimensional problem.
Don’t forget to check your answers using different methods if possible. For example, you could use an online calculator to double check your results.