Edexcel A Level Further Maths: Core Pure:复习笔记6.2.4 Shortest Distances - Planes

How do I find the shortest distance between a given point on a line and a plane?

• The shortest distance from any point on a line to a plane will always be the perpendicular distance from the point to the plane
• Given a point, P, on the line with equation  and a plane  with equation
  • STEP 1: Find the vector equation of the line perpendicular to the plane that goes through the point, P, on
    • This will have the position vector of the point, P, and the direction vector n
  • STEP 2: Find the coordinates of the point of intersection of this new line with  by substituting the equation of the line into the equation of the plane
  • STEP 3: Find the distance between the given point on the line and the point of intersection
    • This will be the shortest distance from the plane to the point
• A question may provide the acute angle between the line and the plane
  • Use right-angled trigonometry to find the perpendicular distance between the point on the line and the plane
    • Drawing a clear diagram will help

How do I find the shortest distance between a plane and a line parallel to the plane?

• The shortest distance between a line and a plane that are parallel to each other will be the perpendicular distance from the line to the plane
• Given a line with equation  and a plane  parallel to with equation
  • Where n is the normal vector to the plane
  • STEP 1: Find the equation of the line perpendicular to and going through the point a in the form
  • STEP 2: Find the point of intersection of the line and
  • STEP 3: Find the distance between the point of intersection and the point,

How do I find the shortest distance between two parallel planes?

• Two parallel planes will never intersect
• The shortest distance between two parallel planes will be the perpendicular distance between them
• Given a plane with equation  and a plane with equation  then the shortest distance between them can be found
  • STEP 1: The equation of the line perpendicular to both planes and through the point a can be written in the form r = a + sn
  • STEP 2: Substitute the equation of the line into  to find the coordinates of the point where the line meets
  • STEP 3: Find the distance between the two points of intersection of the line with the two planes

Consider the parallel planes defined by the equations:

,

.

a) Show that the two planes  and  are parallel.

b) Find the shortest distance between the two planes and .