The statement in question is about the properties of parallel lines in $\mathbb{R}^3$ (the three-dimensional real coordinate space).

(a) Two lines parallel to a third line are parallel.

• True or False?

This statement is false in $\mathbb{R}^3$.

Explanation:

In $\mathbb{R}^2$ (two-dimensional space), the statement would be true because all lines parallel to a given line must be in the same plane, and if two lines are parallel to the same line, they must be parallel to each other.

However, in $\mathbb{R}^3$, two lines can be parallel to a third line but not parallel to each other. This is because, in three-dimensional space, the two lines could lie in different planes. For example, imagine two lines that are both vertical and parallel to the z-axis, but one is on the x-axis and the other is on the y-axis. These two lines are parallel to the z-axis but are not parallel to each other, since they lie in different planes and do not have the same direction.

Thus, the correct answer is False.

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Would you like more details or have any questions?

Related questions:

1. What are the conditions for two lines to be parallel in $\mathbb{R}^3$?
2. Can two skew lines in $\mathbb{R}^3$ be parallel to the same line?
3. How do you determine whether two lines are skew in $\mathbb{R}^3$?
4. What is the difference between parallel and skew lines in $\mathbb{R}^3$?
5. How can two lines be parallel in $\mathbb{R}^2$ but not in $\mathbb{R}^3$?

Tip:

In $\mathbb{R}^3$, when analyzing line relationships, always check whether the lines lie in the same plane or different planes before concluding if they are parallel or skew.