Question

1. Name all the points.
2. Name all the lines using a script letter.
3. What is the other name for "line a"?
4. Using a script letter, name a plane formed by the four lines.
5. Name the plane in three different ways using three points​

Answer 1

Answer:

You can find another name for plane P by using any three points in the plane that are not on the same line. So, another name for plane P is plane FAB. CD , and three collinear points are A, C, and B.

Answer 2

Answer: Two points p1 and p2 define a line if and only if p1≠p2.

Similarly three points define a plane if and only if they are not Collinear (that is they don't all line on a single line).

Finally, four points define a Hyperplane if and only if they are not Coplanar (they don't all line in a single plane).

Step-by-step explanation:

STEP:1 Two points: have a line segment between them. This is a 1-cell(you can think a triangle in one dimension). We reach that line by just expanding this convex hull in both directions. But the real import here, is that hull.

Three points: have three line segments associated to them. If one point lies in the hull of the other two(in between them), you get a degenerate case, which is the previous example. Otherwise, you get a nice little triangle. Fill in that triangle and you'll have a 2-cell. This is the convex hull of 3 points

STEP:2 Just by continuing the pattern, it would go like this:

Given a plane, 2 points define a line within that plane.

Given a volume, 3 points define a plane within that volume.

Given a hypervolume, 4 points define a volume within that hypervolume.

Given an n-Dimensional space, n points define an (n-1)-Dimensional space within it.

If you've worked with local coordinate systems:

2 points (Origin, Axis), define position and orientation of a local coordinate axis.

3 points (Origin, Xaxis, Yaxis) define the position and orientation of a local 2D coordinate system.

4 points (Origin, Xaxis, Yaxis, Zaxis) define the position and orientation of a local 3D coordinate system (also known as a 'World Matrix' to all you video game programmers out there).

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