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

Shortest Distance between a Point & a Plane

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

ZAFnkWOW_3-11-4-ib-hl-aa-shortest-dist-two-planes-diagram-1

Worked Example

The plane  has equation .

The line  has equation .

The point  lies on the line .

Find the shortest distance between the point P and the plane .

3-11-4-ib-hl-aa-shortest-dist-two-planes-we-1

Shortest Distance between a Line & a Plane

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,

3-11-4-ib-hl-aa-shortest-dist-two-planes-diagram-2-1

Shortest Distance between two Planes

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.

al-fm-6-2-2-shortest-distance-between-two-planes-we-solution-a

 

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

al-fm-6-2-2-shortest-distance-between-two-planes-we-solution-b

 

 

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