Answer :
Final answer:
In three-dimensional space, there is exactly one line that can be drawn to join two skew lines L₁ and L₂ such that the line is perpendicular to both skew lines. This unique line is known as the common perpendicular or the line of shortest distance between the skew lines. The answer to the question is that there is exactly one such line that is perpendicular to both L₁ and L₂.
Explanation:
When considering two skew lines L₁ and L₂, that is, lines that do not intersect and are not parallel, there is exactly one line that can be drawn to join them such that this line is perpendicular to both L₁ and L₂. This line is known as the shortest distance between the two skew lines and is often referred to as the common perpendicular or the line of shortest distance.
To understand why there is only one such line, we must consider the properties of three-dimensional space. If we had more than one line that was perpendicular to both L₁ and L₂, we would have two distinct planes defined by each line and one of the skew lines, and these planes would intersect along a line that would then be parallel to both skew lines. However, this contradicts the initial condition that the skew lines are not parallel. Therefore, there can only be one line that is perpendicular to both.
The process to find this common perpendicular involves calculating the direction vectors of each line and then using the cross product to find the direction vector of the perpendicular line. However, this process is not detailed in the question, and instead, we are asked about the number of such lines. The answer to the question is that there is exactly one such line that is perpendicular to both L₁ and L₂.
