What is the shortest distance to plane?

The Shortest Distance to a Plane

The shortest distance between a point and a plane is the perpendicular distance from the point to the plane. This is because the perpendicular line is the shortest path between any two points, and it is also the most direct route between the point and the plane.

In other words, the shortest distance from a point to a plane is the length of the line segment that is perpendicular to the plane and passes through the point. This line segment is called the normal vector to the plane.

The normal vector to a plane is a vector that is perpendicular to every vector in the plane. It can be found by taking the cross product of two vectors that lie in the plane.

Once you have the normal vector to the plane, you can find the shortest distance from a point to the plane by using the following formula:

distance = |(point - plane origin) dot normal vector| / |normal vector|

where:

• point is the point that you are measuring the distance from
• plane origin is a point that lies on the plane
• normal vector is the normal vector to the plane

Here is an example of how to find the shortest distance from a point to a plane:

Problem:

Find the shortest distance from the point (1, 2, 3) to the plane x + y + z = 6.

Solution:

1. Find the normal vector to the plane.

The normal vector to the plane x + y + z = 6 is (1, 1, 1).

2. Find the distance from the point to the plane origin.

The plane origin is the point (0, 0, 0). The distance from the point (1, 2, 3) to the plane origin is:

distance = sqrt((1 - 0)^2 + (2 - 0)^2 + (3 - 0)^2) = sqrt(14)

3. Find the shortest distance from the point to the plane.

The shortest distance from the point (1, 2, 3) to the plane x + y + z = 6 is:

distance = |(1 - 0) * 1 + (2 - 0) * 1 + (3 - 0) * 1| / sqrt(1^2 + 1^2 + 1^2) = sqrt(14) / sqrt(3) ≈ 2.45