1. Why are points and lines hard to define?
A point you can't move at all, a line you can only move back and forth in the same direction.
Does drawing a point or a line accurately illustrate its characteristics?
2. What do we mean when we say a point exists in zero dimensions?
When you are on a point you can't travel at all in any direction while staying on that point. That means you have zero options to travel in. That is why it is said you have zero dimensions.
3. What does normal space refer to?
Normal space refers 3 dimensional space that extends beyond the three dimensions of length, width, and height.
4. Does a sheet of paper accurately represent a plane?
If you can move backward, forwards, up and down in two different directions it is considered two dimensional. The two dimensional figure is considered a plane. For example, if you took a piece of paper that extended forever in every direction, that in a geometric a sense, is a plane.
The piece of paper itself is itself, finite, and you could call the piece of paper a plane segment because it is a segment of an entire plane.
points and lines have no width, thickness, and a point has no length.
Normal space: This is a highly technical definition, I am not certain what your teacher is looking for. Google "Normal Space" for examples.
yes, on segment comment.
To answer the question, "Does drawing a point or a line accurately illustrate its characteristics?" it is important to note that while drawings can provide a visual representation of points and lines, they may not capture their true characteristics fully. Drawings are two-dimensional representations, whereas points and lines exist in higher dimensional spaces. Points are considered to exist in zero dimensions, while lines are considered to exist in one dimension. So, while a drawing on paper can help visualize a point or a line, it cannot fully represent their characteristics in terms of dimensionality.
Furthermore, it is important to understand that the concept of dimensionality extends beyond what can be illustrated or perceived in physical representations. Mathematics and geometry provide formal definitions for points, lines, and other geometric objects based on their properties and relationships.
In summary, while drawings on paper can provide a useful visual aid, they may not accurately depict the full characteristics of points and lines, which exist in higher-dimensional spaces and are defined mathematically.