How to Find the Distance from a Point to a Plane?

Finding the distance from a point to a plane is a common problem in geometry and can be solved using a straightforward formula.

Understanding the Problem

Imagine you have a point $P(x_1, y_1, z_1)$ and a plane defined by the equation $Ax + By + Cz + D = 0$. The goal is to determine the shortest distance from the point to this plane.

The Formula

The distance $d$ from the point $P(x_1, y_1, z_1)$ to the plane $Ax + By + Cz + D = 0$ can be calculated using the following formula:

$d = frac{|Ax_1 + By_1 + Cz_1 + D|}{sqrt{A^2 + B^2 + C^2}}$

Breaking Down the Formula

1. Numerator: This is the absolute value of the plane equation when the point’s coordinates are substituted. It represents the perpendicular distance without considering the direction.
2. Denominator: This is the magnitude of the normal vector to the plane, which is derived from the coefficients $A$, $B$, and $C$

Step-by-Step Example

Let’s use an example to make this clearer. Suppose you have a point $P(1, 2, 3)$ and a plane defined by the equation $2x + 3y – z + 4 = 0$

1. Substitute the point into the plane equation:

$2(1) + 3(2) – 1(3) + 4 = 2 + 6 – 3 + 4 = 9$

2. Take the absolute value:

$|9| = 9$

3. Calculate the magnitude of the normal vector:

$sqrt{2^2 + 3^2 + (-1)^2} = sqrt{4 + 9 + 1} = sqrt{14}$

4. Divide the absolute value by the magnitude:

$d = frac{9}{sqrt{14}}$

5. Simplify the fraction if necessary:

$d = frac{9sqrt{14}}{14}$

So, the distance from the point $(1, 2, 3)$ to the plane $2x + 3y – z + 4 = 0$ is $frac{9sqrt{14}}{14}$ units.

Conclusion

Understanding how to find the distance from a point to a plane is crucial for various applications in geometry, physics, and engineering. By substituting the point’s coordinates into the plane equation and following the formula, you can easily determine this distance. Practice with different points and planes to become more comfortable with the process.

Citations

1. 1. Khan Academy – Distance from a point to a plane
2. 2. Paul’s Online Math Notes – Distance from a Point to a Plane
3. 3. MIT OpenCourseWare – Distance from a Point to a Plane
4. 4. Math Insight – Distance from a point to a plane