How can you determine the equation of a plane given three non-collinear points in space?

Question

This question encourages discussion about fundamental concepts in geometry and can lead to an exploration of how planes are mathematically defined.

Answer 1

Call the points P, Q, and R.

v = PQ and w = PR are vectors parallel to the plane

n = v x w is perpendicular to the plane

If P = (a, b, c) and (x, y, z) is an arbitrary point in the plane, then

<x-a, y-b, z-c> is a vector that is both parallel to the plane and perpendicular to n.

Equation of the plane: n • < x-a, y-b, z-c> = 0

Answer 2

Here is a URL to the Desmos graph used in the video:

desmos.com/3d/3styxnvmcu

Answer 3

To find the equation of a plane when you have three points (let's call them A, B, and C):

1. Visualize the Plane: Imagine the three points A, B, and C form a flat surface in space. This surface is what we're calling the plane.
2. Create the Flat Surface: You can use the positions of these points to describe this surface with a simple formula. This formula tells us everything we need to know about the plane’s orientation and position.
3. Use a Formula: By plugging the coordinates of points A, B, and C into a specific formula, you get the equation of the plane. It’s like finding the recipe for making a flat sheet that perfectly fits through those points.

In essence, finding the equation of the plane involves using the three points to define exactly where this flat surface is in space. It's a neat way of seeing how these points all lie on the same plane!