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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesCoincident, Parallel, Intersecting & Skew Lines

Coincident, Parallel, Intersecting & Skew Lines (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

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DP Analysis & Approaches (AA): HLDP Analysis & Approaches (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 their direction vectors are scalar multiples

  • e.g. 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 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 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 lines that lie directly on top of each other

    • They are the same line

  • Two lines are coincident if:

    • Their direction vectors are scalar multiples

    • Any point on one of the lines also lies on the other

  • e.g. 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

    • 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 so the direction vectors are scalar multiples

    • begin mathsize 16px style open parentheses table row 1 row cell negative 8 end cell end table close parentheses end styleis a position vector of a point on the first line

    • 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

What are skew lines?

  • Two lines are skew if:

    • They are not parallel

    • They do not intersect

  • Skew lines are only possible in 3-dimensions

How do I determine whether two non-parallel lines in 3 dimensions are skew or intersecting?

  • STEP 1
    Set the vector equations of the two lines equal to each other with different variables

    • e.g. use lambda and mu for the parameters

  • STEP 2
    Write the three separate equations for the i, j, and k components in terms of lambda and mu

  • STEP 3
    Solve two of the equations to find values for lambda and mu

  • STEP 4
    Check whether the values of lambda and mu 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

Examiner Tips and Tricks

If you try to do this with parallel lines, then your equations will not have a unique solution. If the lines are coincident, then your equations will result in a statement that is always true, such as 0 equals 0. If they are parallel but not coincident, then the equations will result in a statement that is never true such as 1 equals 0.

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

  • Follow the steps above to find the values of lambda and mu that satisfy all three equations

  • Find the point of intersection by substituting either the value of lambda or the value of mu into one of the vector equations

Examiner Tips and Tricks

It is always a good idea to check that you get the same position vector if you substitute the other value into the other equation.

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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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.