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
- STEP 1: Find the vector equation of the line perpendicular to the plane that goes through the point, P, on
- 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
- Use right-angled trigonometry to find the perpendicular distance between the point on the line and the plane
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 .