Explain the possible relationships between two lines in R^3

take two rulers and wave them through the space in front of you.

How many different situations can you create ?

In R³ (three-dimensional space), there are three possible relationships between two lines: they can be parallel, intersecting, or skew. Let's discuss how to determine the relationship between two lines in R³.

To determine if two lines are parallel, we need to compare their direction vectors. The direction vector of a line is obtained by subtracting the coordinates of any two points on the line. If the direction vectors of the two lines are scalar multiples of each other, then the lines are parallel. Mathematically, we can write it as:

Line 1: r₁ = a₁ + λd₁
Line 2: r₂ = a₂ + μd₂

where r₁ and r₂ are points on Line 1 and Line 2, a₁ and a₂ are their respective position vectors, and d₁ and d₂ are the direction vectors. If d₁ = k * d₂ (k is a constant), then the lines are parallel.

If the direction vectors are not scalar multiples of each other, the lines might intersect. To determine if they do intersect, we need to find a point that lies on both lines. We can set the position vector expressions for the two lines equal to each other:

a₁ + λd₁ = a₂ + μd₂

We can then solve this system of equations to find the values of λ and μ. If there is a solution where both λ and μ are real numbers, then the lines intersect at that point.

Finally, if there is no scalar relationship between the direction vectors and the lines do not intersect, then the lines are skew. Skew lines are lines that do not lie in the same plane and never intersect.

To summarize:
- If the direction vectors are scalar multiples of each other, the lines are parallel.
- If the lines intersect at a point, but their direction vectors are not scalar multiples, the lines are intersecting.
- If the lines do not have a scalar relationship between their direction vectors and never intersect, the lines are skew.