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
  • For example, the lines whose equations are and  are parallel
    • This is because 

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
    • for example, the lines represented by the equations  and  are coincident,
  • To check whether two lines are coincident:
    • First check that they are parallel
      • They are because  and so their direction vectors are parallel     
    • Next, determine whether any point on one of the lines also lies on the other
      • is the position vector of a point on the first line and  so it also lies on the second line
    • 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 of λ 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

Worked Example

How do we find the angle between two lines?

• The angle between two lines is equal to the angle between their direction vectors
  • It can be found using the scalar product of their direction vectors
• Given two lines in the form  and  use the formula
  • 
• If you are given the equations of the lines in a different form or two points on a line you will need to find their direction vectors first
• To find the angle ABC the vectors BA and BC would be used, both starting from the point B
• The intersection of two lines will always create two angles, an acute one and an obtuse one
  • These two angles will add to 180°
  • You may need to subtract your answer from 180° to find the angle you are looking for
  • A positive scalar product will result in the acute angle and a negative scalar product will result in the obtuse angle
    • Using the absolute value of the scalar product will always result in the acute angle

Worked Example