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
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 .