Finding the Closest Point on a Plane to a Given Point

In summary, the problem is to find the point on the plane x-y+z=4 that is closest to the point (1,2,3). The shortest distance from the origin to any plane with the equation ax+by+cz=d is given by d=2/sqrt(3). To find the point on the given plane at this distance from (1,2,3), we can embed the point into a new plane with the same normal vector and calculate the difference.
  • #1
Char. Limit
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Homework Statement


Find the point on the plane [itex]x-y+z=4[/itex] closest to the point (1,2,3).

[tex]d=\frac{2}{\sqrt{3}}[/tex]

Homework Equations



Hmm...

The Attempt at a Solution



As you can see, I've already solved for the shortest distance. But, knowing the plane I'm on and the distance, how do I find the point that lies on that plane, at that distance from (1,2,3)? Help...
 
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  • #2
Embed the point into a plane with the same normal vector as the given plane.

The shortest distance from the origin to any plane [tex] ax +by +cz = d [/tex] is
[tex] \frac{|\mathbf{n}|}{d} [/tex] where [tex] \mathbf{n} [/tex] is the normal vector to the plane.

Do this for both the given plane and your new plane from and calculate the difference.
 

What is the concept of minimum distance from a plane?

The minimum distance from a plane is the shortest distance between a point and a plane in three-dimensional space. It is the perpendicular distance from the point to the plane, measured along a line that is perpendicular to the plane.

How is the minimum distance from a plane calculated?

The minimum distance from a plane can be calculated using the formula d = |ax0 + by0 + cz0 + d| / √(a2 + b2 + c2), where (x0, y0, z0) is the coordinates of the point and ax + by + cz + d = 0 is the equation of the plane in standard form.

Why is minimum distance from a plane important in science?

The concept of minimum distance from a plane is important in various fields of science such as physics, engineering, and mathematics. It is used to calculate the distance between an object and a plane, which is essential in analyzing the behavior of particles, designing structures, and solving geometric problems.

What are some real-life applications of minimum distance from a plane?

Minimum distance from a plane has numerous real-life applications, including navigation systems, collision avoidance systems, aircraft design, and satellite imaging. It is also used in industries such as architecture, construction, and computer graphics.

Can the minimum distance from a plane be negative?

No, the minimum distance from a plane cannot be negative. It is always a positive value, as it represents the shortest distance between a point and a plane. A negative value would indicate that the point is on the opposite side of the plane from the origin.

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