Lines in Space

Example: Determine whether the two lines L₁ and L₂ are parallel.

L₁ x = 6 + 2t, y = 5 + 2t, z = 7 + 3t

L₂ x = 7 + 3s, y = 5 + 3s, z = 10 + 5s

Note the lines in “symmetric” form are:

(x - 6)/2 = (y – 5)/2 = (z – 7)/3 = t

and (x – 7)/3 = (y – 5)/ 3 = (z – 10)/5 = s

So, the vector parallel to line L₁ is V₁ = 2 i + 2 j + 3 k

and the vector parallel to line L₂ is V₂ = 3 i + 3 j + 5 k

Since V₁ is not a multiple of V₂ , these vectors and the lines are not parallel. (result)

Next determine if the lines L₁ and L₂ skew or intersecting.

If the lines intersect, then they must have a common point at the point of intersection.

If that is not the case, then the lines are skew.

So assume that the two lines intersect. Then:

x = 6 + 2t = 7 + 3s eq (1)

y = 5 + 2t = 5 + 3s eq (2)

z = 7 + 3t = 10 + 5s eq (3)

Subtract (2) from (3). The result is: 2 + t = 5 + 2s or t = 3 + 2s

So 2t = 6 + 4s . Put this into (2) and solve for s.

5 + 6 + 4s = 5 + 3s, or s = -6

Then put this result into t = 3 + 2s which yields t = -9

Now determine if these values of s and t satisfy eq (1).

6 + 2(-9) =? 7 + 3(-6) which does not hold!

So the lines L₁ and L₂ do not intersect. Therefore they are skew. (result)

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