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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesShortest Distance Between Two Lines

Shortest Distance Between Two Lines (DP IB Analysis & Approaches (AA)): Revision Note

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DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

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Shortest Distance Between Two Lines

How do we find the shortest distance between two parallel lines?

  • Two parallel lines will never intersect

  • The shortest distance between two parallel lines will be the perpendicular distance between them

  • Given a line l subscript 1 with equation bold r equals bold a subscript 1 plus lambda bold d subscript 1and a line begin mathsize 16px style l subscript 2 end style with equation bold r equals bold a subscript 2 plus mu bold d subscript 2 then the shortest distance between them can be found using the following steps:

    • STEP 1: Find the vector between bold a subscript 1 and a general coordinate from l subscript 2 in terms of μ  

    • STEP 2: Set the scalar product of the vector found in STEP 1 and the direction vector Error converting from MathML to accessible text.equal to zero

      • Remember the direction vectors bold d subscript 1 and are scalar multiples of each other and so either can be used here

    • STEP 3: Form and solve an equation to find the value of μ

    • STEP 4: Substitute the value of μ  back into the equation for l subscript 2 to find the coordinate on l subscript 2 closest to l subscript 1

    • STEP 5: Find the distance between bold a subscript 1 and the coordinate found in STEP 4

  • Alternatively, the formula fraction numerator open vertical bar stack A B with rightwards arrow on top cross times bold d close vertical bar blank over denominator open vertical bar bold d close vertical bar end fraction can be used

    • Where stack A B with rightwards arrow on top is the vector connecting the two given coordinates and bold a subscript 2  

    • d is the simplified vector in the direction of bold d subscript 1 and bold d subscript 2

    • This is not given in the formula booklet

How do we find the shortest distance from a given point on a line to another line?

  • The shortest distance from any point on a line to another line will be the perpendicular distance from the point to the line

  • If the angle between the two lines is known or can be found then right-angled trigonometry can be used to find the perpendicular distance

    • The formula fraction numerator open vertical bar stack A B with rightwards arrow on top cross times bold d close vertical bar blank over denominator open vertical bar bold d close vertical bar end fraction given above is derived using this method and can be used

  • Alternatively, the equation of the line can be used to find a general coordinate and the steps above can be followed to find the shortest distance

How do we find the shortest distance between two skew lines?

  • Two skew lines are not parallel but will never intersect

  • The shortest distance between two skew lines will be perpendicular to both of the lines

    • This will be at the point where the two lines pass each other with the perpendicular distance where the point of intersection would be

    • The vector product of the two direction vectors can be used to find a vector in the direction of the shortest distance

    • The shortest distance will be a vector parallel to the vector product

  • To find the shortest distance between two skew lines with equations bold r equals bold a subscript 1 plus lambda bold d subscript 1 and bold r equals bold a subscript 2 plus mu bold d subscript 2 ,

    • STEP 1: Find the vector product of the direction vectors bold space bold d subscript 1 and bold space bold d subscript 2

      • bold d bold space equals blank bold d subscript 1 blank cross times blank bold d subscript 2

    • STEP 2: Find the vector in the direction of the line between the two general points on l subscript 1 and l subscript 2  in terms of λ  and μ

      • stack A B with rightwards arrow on top space equals blank bold b blank minus blank bold a blank

    • STEP 3: Set the two vectors parallel to each other

      • k bold d bold space equals space stack A B with rightwards arrow on top

    • STEP 4: Set up and solve a system of linear equations in the three unknowns, k comma blank lambda and mu

3-10-5-ib-aa-hl-short-distance-lines-diagram-1

Examiner Tips and Tricks

  • Exam questions will often ask for the shortest, or minimum, distance within vector questions

  • If you’re unsure start by sketching a quick diagram

  • Sometimes calculus can be used, however usually vector methods are required

Worked Example

Consider the skew lines l subscript 1 and l subscript 2 as defined by:

l subscript 1: bold italic r equals open parentheses table row 6 row cell negative 4 end cell row 3 end table close parentheses plus lambda open parentheses table row 2 row cell negative 3 end cell row cell blank 4 blank end cell end table close parentheses

 

l subscript 2: bold italic r equals open parentheses table row cell negative 5 end cell row 4 row cell negative 8 end cell end table close parentheses plus mu open parentheses table row cell negative 1 end cell row 2 row cell blank 1 blank end cell end table close parentheses

Find the minimum distance between the two lines.

3-10-5-ib-aa-hl-short-distance-lines-we-2

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    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.