IB DP Maths: AI HL复习笔记3.8.2 Shortest Distances with Lines

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

• Note that the shortest distance between the point and the line is sometimes referred to as the length of the perpendicular

How do we use the vector product to find the shortest distance from a point to a line?

• The vector product can be used to find the shortest distance from any point to a line on a 2-dimensional plane

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

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
  • 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