How can 3 noncollinear points determine a plane
To understand how three noncollinear points determine a plane, it is helpful to first define what we mean by collinear and noncollinear points.
Collinear points are points that lie on the same straight line. In other words, if you can draw a straight line that passes through multiple points, those points are collinear.
Noncollinear points, on the other hand, are points that do not lie on the same line. If you cannot draw a straight line that passes through all the points, they are considered noncollinear.
Now, let's consider how three noncollinear points can determine a plane in three-dimensional space.
In three-dimensional space, a plane is a flat, two-dimensional surface that extends infinitely in all directions. It is determined by three noncollinear points.
To understand why three noncollinear points are sufficient, we can use the following reasoning:
1. Start with the first point and imagine a line passing through it.
2. Next, take the second point and align it with the first point.
3. Now, imagine a line passing through both the first and second points.
4. Finally, take the third point and align it with the first two points.
5. By imagining a line passing through the third point as well, we see that all three lines intersect at a single point.
The intersection point of these three lines represents the location where all three lines meet and marks a unique spot in space. This spot is the origin of a plane.
Since a plane extends infinitely in all directions, it can be imagined as an infinite flat surface passing through all three noncollinear points. Any additional points within this plane can be reached by extending the same idea of intersecting lines through different combinations of the three original points.
In summary, three noncollinear points uniquely determine a plane in three-dimensional space by establishing a common meeting point where three lines intersect.