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 so