Shortest distance from a point to a plane

Shortest Distance from a Point to a Plane

Understanding the Basics

  • A plane in three-dimensional space can be defined by its scalar equation, which is the dot product of the normal vector and the position vector, equalling a constant.
  • The normal vector is perpendicular to the plane.
  • A point in three-dimensional space is defined by its position vector.

The Scalar Equation of a Plane

  • The scalar equation of a plane is given by n.r = p, where n is the normal of the plane, r is the position vector of any point on the plane and p is a constant.
  • Understanding the scalar equation is crucial, as it helps in determining the shortest distance from a point to the plane.

Finding the Shortest Distance

  • The shortest distance from a point to a plane is the perpendicular distance from that point to the plane.
  • To compute this distance, you need the position vectors of the point and of any point on the plane, as well as the normal to the plane.

The Formula for the Shortest Distance

  • The formula for the shortest distance d from a point A to a plane is given by **d = n.(a − p) ÷ n **, where:
    • n is the normal to the plane,
    • a is the position vector of point A,
    • p is the position vector of any point P on the plane,
    • the dot ”.” represents the dot product,
    • the absolute value **” “** indicates the magnitude of the vector or dot product.

Calculating the Shortest Distance

  • To find the shortest distance:
    1. Obtain the position vector of the point, the position vector of any point on the plane and the plane’s normal vector.
    2. Subtract the point on the plane’s position vector from the position vector of the point to create a new vector.
    3. Compute the dot product between the normal vector and the new vector from step (2).
    4. Take the absolute value of the dot product from step (3).
    5. Divide this absolute value by the magnitude of the normal vector.

Remember, the shortest distance from a point to a plane in three-dimensional space is a very important concept in Further Pure Mathematics. It is essential to understand and remember both the formula and the steps to compute it.