Calculating the Distance from a Point to a Plane Using the Perpendicular Distance Formula

In three-dimensional geometry, the distance of a point from a plane is an essential concept used in various applications, including computer graphics, engineering design, and advanced mathematics. This article will guide you through the process of finding the perpendicular distance from a given point to a plane using the distance formula.

Understanding the Concept

The distance ( D ) from a point ( (x_1, y_1, z_1) ) to a plane given by the equation ( ax by cz d 0 ) is the length of the perpendicular line segment from the point to the plane. This distance can be calculated using a specific formula derived from the Pythagorean theorem.

The Formula

Given a plane with the equation:

[ ax by cz d 0 ]

The distance ( D ) from a point ( (x_1, y_1, z_1) ) to this plane is given by:

[ D frac{|ax_1 by_1 cz_1 d|}{sqrt{a^2 b^2 c^2}} ]

Step-by-Step Calculation

Let's apply this formula to a specific example. We are given a point ( (3, -2, 1) ) and a plane ( 2x - y 2z 3 0 ). To find the distance from the point to the plane, follow these steps:

Identify the coordinates of the point: Point ( (x_1, y_1, z_1) (3, -2, 1) ) Identify the coefficients of the plane equation: ( a 2, b -1, c 2, d 3 ) Substitute these values into the formula: [ D frac{|2(3) (-1)(-2) 2(1) 3|}{sqrt{2^2 (-1)^2 2^2}} ]

Performing the Calculation

Let's break down the calculation step by step:

Calculate the numerator:

[ 2(3) (-1)(-2) 2(1) 3 6 2 2 3 13 ]

Calculate the denominator:

[ sqrt{2^2 (-1)^2 2^2} sqrt{4 1 4} sqrt{9} 3 ]

Compute the distance:

[ D frac{13}{3} ]

Conclusion

The distance from the point ( (3, -2, 1) ) to the plane ( 2x - y 2z 3 0 ) is ( frac{13}{3} ).

This method allows you to calculate the distance between any point and plane, as long as the equation of the plane and the coordinates of the point are known. It is a fundamental concept in three-dimensional geometry with numerous practical applications.