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 .