91Ó°ÊÓ

When dealing with geometry in three-dimensional space, understanding how to calculate the distance between a point and a plane is crucial.
The formula used to find this distance can look complex, but it's quite manageable when broken down. The general distance formula for a point \((x_1, y_1, z_1)\) to a plane described by the equation \(Ax + By + Cz + D = 0\) is:
\[ Distance = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
This formula calculates the shortest distance from a point to a plane, which is always along a line perpendicular to the plane.
To use this formula, you'll need to identify the coefficients of the plane equation and the coordinates of the point precisely. With these values, substituting into the formula will give you the required distance.

The plane equation is a mathematical representation of a flat, two-dimensional surface in three-dimensional space.
Generally, it takes the form \(Ax + By + Cz + D = 0\). Each term in the equation serves a specific purpose:
- 'A', 'B', and 'C' are coefficients that determine the plane's orientation in space.
- 'x', 'y', and 'z' are variables that represent any point lying on the plane.
- 'D' is a constant that shifts the plane within the space.
In the example provided, the plane equation is \(x - y + z = 4\).
This translates to \(1x - 1y + 1z - 4 = 0\).
Identifying these components correctly allows us to utilize them in calculations such as finding the distance to a point.

A point in three-dimensional space is defined by its three coordinates: \((x, y, z)\).
These coordinates tell us exactly where the point is located in relation to the origin of the coordinate system.
Knowing the exact coordinates is essential for various computations, including measuring distance.
For instance, consider a point given as \((2, -1, -1)\):
- 'x-coordinate' or \(x_1 = 2\)
- 'y-coordinate' or \(y_1 = -1\)
- 'z-coordinate' or \(z_1 = -1\)
Accurately identifying these values ensures that we get precise results when substituting into formulas or equations. Remember, tiny errors in coordinates can lead to incorrect outcomes.