Shortest Distance Between Two Lines (DP IB Applications & Interpretation (AI)): Revision Note

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  with equation and a line  with equation then the shortest distance between them can be found using the following steps:

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

  • STEP 2: Set the scalar product of the vector found in STEP 1 and the direction vector equal to zero

    • Remember the direction vectors 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 to find the coordinate on closest to 

  • STEP 5: Find the distance between and the coordinate found in STEP 4

• Alternatively, the formula  can be used

  • Where  is the vector connecting the two given coordinates and  

  • d is the simplified vector in the direction of  and 

  • 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  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  and ,

  • STEP 1: Find the vector product of the direction vectors and

    • 

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

    • 

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

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  • STEP 4: Set up and solve a system of linear equations in the three unknowns, and