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

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 =