Edexcel A Level Further Maths: Core Pure:复习笔记6.1.4 Shortest Distances - Lines

Shortest Distance between a Point & 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   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  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  in terms of λ
    • STEP 5: The scalar product of the direction vector of the line l and the displacement vector  will be zero
      • Form an equation  and solve to find λ
    • STEP 6: Substitute λ into  and find the magnitude
      • The shortest distance from the point to the line will be the magnitude of
  • 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

Exam Tip

  • 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  has equation .

Find the shortest distance from A to the line .

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

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:
    • Remember the direction vectors  and  are scalar multiples of each other and so either can be used here
    • 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
    • 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

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
  • 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
  • To find the shortest distance between two skew lines with equations and  ,
    • STEP 1: Find position vectors for the points on each line that form the shortest distance
      • Point P has position vector
      • Point Q has position vector
    • STEP 2: Find the displacement vector between P and Q
    • STEP 3: Form two equations by using the fact that the scalar product of the displacement vector and the direction vector of each line should equal zero
    • STEP 4: Solve the two equations simultaneously to find the values of λ and μ
    • STEP 5: Substitute the values of λ and μ into the displacement vector and take the magnitude
      • Shortest distance =

Exam Tip

  • 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 and  as defined by:

 

Find the minimum distance between the two lines.

al-fm-6-1-4-shortest-distance-between-two-lines-we-solution

 

 

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