In two dimensions, lines are either parallel or they intersect at a single point. If they are parallel, then either they have no points in common, or they share every point in common.

In three dimensions, there is a further possibility: a pair of lines might not be parallel and have no points of intersection. We say that the lines are?skew.

How do I tell if two lines are parallel?

• Two lines are parallel if, and only if, their?direction vectors?are?parallel
  • This means the direction vectors will be?scalar multiples?of each other
  • If two parallel lines share any point, then they share?all points?– i.e. they are?identical

What are skew lines?

• First, start with another question: do lines which are not parallel necessarily intersect?
  • In?2 dimensions, the answer is?yes
  • However, lines in?3 dimensions?do not necessarily intersect

• Lines that are?not parallel?and which?do not intersect?are called?skew lines

How do I determine whether lines in 3D are parallel, skew, or intersecting?

• First, look to see if the direction vectors are parallel:
  • if the?direction vectors are parallel, then the?lines are parallel
  • if the?direction vectors are not parallel, the?lines are not parallel

• If the lines are?parallel, check to see if they are?identical:
  • If they?share any point, then they are?identical
  • If?any point?on one line is?not on the other line, then the lines are?not identical

• If the lines are?not parallel, check whether they intersect:
  • Using?different letters,?e.g.?s?and?t, for the parameters, write down coordinates for a general point on each line
  • Supposing that the lines do intersect: equate the two coordinates and write down?three equations
    • One for each component (i, j, k)
  • Solve?any two pairs?of these equations simultaneously to find?s?and?t
  • Check?whether the values of?s?and?t?you have found satisfy the third equation
    • If?all three?equations are satisfied, then the lines?intersect
    • If?not all three?equations are satisfied, then the lines are?skew

• If a pair of lines are?not parallel?and?do intersect, the unique point of intersection can be found by substituting the value of one of the parameters you have found into the coordinates for points on the appropriate line.