IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesShortest Distances with Lines

Shortest Distances with Lines (DP IB Applications & Interpretation (AI)) : Revision Note

Amber

Author

Amber

Last updated

DP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HL

Did this video help you?

Shortest Distance Between a Point and a Line

How do I find the shortest distance from a point to a line?

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

    • Given a line l  with equation bold r bold space equals bold space bold a plus straight lambda bold b  and a point P not on l

    • The scalar product of the direction vector, b, and the vector in the direction of the shortest distance will be zero

  • The shortest distance can be found using the following steps:

    • STEP 1: Let the vector equation of the line be r and the point not on the line be P, then the point on the line closest to P will be the point F

      • The point F is sometimes called the foot of the perpendicular

    • STEP 2: Sketch a diagram showing the line l and the points P and F

      • The vector stack F P with rightwards arrow on top will be perpendicular to the line l

    • STEP 3: Use the equation of the line to find the position vector of the point F  in terms of λ

    • STEP 4: Use this to find the displacement vector stack F P with rightwards arrow on top in terms of λ

    • STEP 5: The scalar product of the direction vector of the line l and the displacement vector stack F P with rightwards arrow on top will be zero

      • Form an equation stack F P with rightwards arrow on top times bold b equals 0 and solve to find λ

    • STEP 6: Substitute λ into stack F P with rightwards arrow on top and find the magnitude open vertical bar stack F P with rightwards arrow on top close vertical bar 

      • The shortest distance from the point to the line will be the magnitude of stack F P with rightwards arrow on top

  • Note that the shortest distance between the point and the line is sometimes referred to as the length of the perpendicular

7-3-4-foot-of-the-perpendicular

How do we use the vector product to find the shortest distance from a point to a line?

  • The vector product can be used to find the shortest distance from any point to a line on a 2-dimensional plane

  • Given a point, P, and a line r = a + λb

    • The shortest distance from P to the line will befraction numerator open vertical bar stack A P with rightwards arrow on top cross times b close vertical bar blank over denominator open vertical bar b close vertical bar end fraction

    • Where A is a point on the line

    • This is not given in the formula booklet

Examiner Tips and Tricks

  • Column vectors can be easier and clearer to work with when dealing with scalar products.

Worked Example

Point A  has coordinates (1, 2, 0) and the line l has equation bold r equals open parentheses table row 2 row 0 row 6 end table close parentheses plus lambda open parentheses table row 0 row 1 row 2 end table close parentheses. 

Point B lies on the l such that open square brackets A B close square brackets  is perpendicular to l.

Find the shortest distance from A to the line l.

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

Did this video help you?

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 begin mathsize 16px style l subscript 1 end style 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

      • bold d bold space equals space k 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

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 vector methods are usually required

Worked Example

A drone travels in a straight line and at a constant speed. It moves from an initial point (-5, 4, -8) in the direction of the vector open parentheses negative 1 space space 2 space space 1 close parentheses. At the same time as the drone begins moving a bird takes off from initial point (6, -4, 3) and moves in a straight line at a constant speed in the direction of the vector open parentheses space space 2 minus 3 space space 4 close parentheses.

Find the minimum distance between the bird and the drone during this movement.

3-10-5-ib-aa-hl-short-distance-lines-we-2
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Test yourselfFlashcards

Did this page help you?

Previous:Vector Equations of LinesNext:Kinematics with Vectors
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.

Download notes on Shortest Distances with Lines