IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.10 Vector Equations of Lines3.10.3 Pairs of Lines in 3D

Pairs of Lines in 3D (DP IB Maths: AA HL)

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Coincident, Parallel, Intersecting & Skew 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

 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

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

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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Angle Between Two Lines

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 bold italic r equals bold italic a subscript 1 plus lambda bold italic b subscript 1 and bold italic r equals bold italic a subscript 2 plus lambda bold italic b subscript 2 use the formula
    • begin mathsize 16px style theta equals cos to the power of negative 1 end exponent invisible function application open parentheses fraction numerator bold italic b subscript 1 blank bullet blank bold italic b subscript 2 over denominator open vertical bar bold italic b subscript 1 close vertical bar open vertical bar blank bold italic b subscript 2 close vertical bar end fraction close parentheses end style
  • 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

Examiner Tip

  • In your exam read the question carefully to see if you need to find the acute or obtuse angle
    • When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question

Worked example

Find the acute angle, in radians between the two lines defined by the equations:

begin mathsize 16px style l subscript 1 colon space space bold italic a equals open parentheses table row 2 row 0 row cell blank 3 blank end cell end table close parentheses plus lambda open parentheses table row 1 row cell negative 4 end cell row cell blank minus 3 blank end cell end table close parentheses end style and  begin mathsize 16px style l subscript 2 colon space space bold italic b equals open parentheses table row 1 row cell negative 4 end cell row 3 end table close parentheses plus mu open parentheses table row cell blank minus 3 blank end cell row 2 row 5 end table close parentheses end style

R_UZJlZ8_3-10-3-ib-aa-hl-angle-between-we-solution-2a

 

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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.