IB DP Maths: AA HL复习笔记3.10.3 Pairs of Lines in 3D

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