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A LevelFurther Maths: Core PureEdexcelRevision Notes6. Further Vectors6.1 Vector Lines6.1.2 Pairs of Lines in 3D

Pairs of Lines in 3D (Edexcel A Level Further Maths: Core Pure)

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Coincident & Parallel Lines

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 begin mathsize 16px style bold r equals open parentheses table row 2 row 1 row cell negative 7 end cell end table close parentheses plus lambda subscript 1 open parentheses table row 2 row 0 row cell negative 8 end cell end table close parentheses blank end styleand begin mathsize 16px style bold r equals open parentheses table row 1 row cell negative 1 end cell row 5 end table close parentheses plus lambda subscript 2 open parentheses table row cell negative 1 end cell row 0 row 4 end table close parentheses end style are parallel
      • This is because begin mathsize 16px style open parentheses table row 2 row 0 row cell negative 8 end cell end table close parentheses equals negative 2 open parentheses table row cell negative 1 end cell row 0 row 4 end table close parentheses end style

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 bold r equals open parentheses table row 1 row cell negative 8 end cell end table close parentheses plus s open parentheses table row cell negative 4 end cell row 8 end table close parentheses and bold r equals open parentheses table row cell negative 3 end cell row 0 end table close parentheses plus t open parentheses table row 1 row cell negative 2 end cell end table close parentheses are coincident,
    • To check whether two lines are coincident:
      • First check that they are parallel
        • They are because begin mathsize 16px style open parentheses table row cell negative 4 end cell row 8 end table close parentheses equals negative 4 open parentheses table row 1 row cell negative 2 end cell end table close parentheses end style and so their direction vectors are parallel     
      • Next, determine whether any point on one of the lines also lies on the other
        • begin mathsize 16px style open parentheses table row 1 row cell negative 8 end cell end table close parentheses end styleis the position vector of a point on the first line and begin mathsize 16px style open parentheses table row 1 row cell negative 8 end cell end table close parentheses equals open parentheses table row cell negative 3 end cell row 0 end table close parentheses plus 4 open parentheses table row 1 row cell negative 2 end cell end table close parentheses end style so it also lies on the second line
      • If two parallel lines share any point, then they share all points and are coincident

Intersecting Lines

How do I determine whether two lines in 3 dimensions intersect?

  • 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

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

Examiner Tip

  • Make sure that you use different letters, e.g. lambda and mu, to represent the parameters in vector equations of different lines
    • Check that the variable you are using has not already been used in the question

Worked example

Line L1 has vector equation bold r equals open parentheses table row 8 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses plus lambda open parentheses table row 3 row cell negative 1 end cell row 1 end table close parentheses.

Line L2 has vector equation bold r equals open parentheses table row cell negative 3 end cell row 11 row 2 end table close parentheses plus mu open parentheses table row cell negative 1 end cell row 2 row 1 end table close parentheses.

a)
Show that the lines L1 and L2 intersect.

al-fm-6-1-2-intersecting-lines-we-solution-a

 

b)
Find the position vector of the point of intersection.

al-fm-6-1-2-intersecting-lines-we-solution-b

Skew Lines

 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
  • If two lines are skew then there is not a plane in 3D than contains both of the lines

7-3-2-parallel-intersecting-_-skew-lines

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

 

Worked example

Determine whether the following pair of lines are parallel, intersect, or are skew.

bold r equals 4 bold i plus 3 bold j plus s open parentheses 5 bold i plus 2 bold j plus 3 bold k close parentheses and bold italic r equals negative 5 bold i plus 4 bold j plus bold k plus t open parentheses 2 bold i minus bold j close parentheses.

JY6QiVwy_3-10-3-ib-aa-hl-angle-between-we-solution-1

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.