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
  • How do I tell if two lines are coincident?

• Coincident lines?are two lines that lie directly on top of each other
  • They are indistinguishable from each other
• Two parallel lines will either?never intersect?or they are?coincident (identical)
  • Sometimes the vector equations of the lines may look different
    • 
  • • If two parallel lines share?any point, then they share?all points?and are?coincident

?What are skew lines?

• Lines that are?not parallel?and which?do not intersect?are called?skew lines
  • This is only possible in?3-dimensions

How do I determine whether lines in 3 dimensions 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 the lines are?coincident:
  • If they?share any point, then they are?coincident
  • If?any point?on one line is?not on the other line, then the lines are?not coincident
• If the lines are?not parallel, check whether they?intersect:
  • STEP 1: Set the vector equations of the two lines equal to each other with?different variables
    • e.g. λ and μ, for the parameters
  • STEP 2: Write the three separate equations for the?i,?j, and?k?components in terms ofλ and μ
  • STEP 3:?Solve?two of the equations to find a value for λ and μ
  • STEP 4:?Check?whether the values of λ and μ 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

How do I find the point of intersection of two lines?

• If a pair of lines are?not parallel?and?do intersect, a unique point of intersection can be found
  • If the two lines intersect, there will be a single point that will lie on both lines
• Follow the steps above to find the values of λ and μ that satisfy?all three equations
  • STEP 5: Substitute either the value of λ or the value of μ into one of the vector equations to find the position vector of the point where the lines intersect
  • It is always a good idea to?check?in the other equations as well, you should get the same point for each line