Point on plane closest to point in R3

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    Plane Point
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Homework Help Overview

The problem involves finding the point on a specified plane that is closest to a given point in three-dimensional space. The plane is defined as passing through a specific point and is parallel to the xy-plane.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss deriving the equation of the plane and the concept of distance from a point to a plane, particularly focusing on the orthogonal relationship.

Discussion Status

The discussion includes attempts to clarify the relationship between the point and the plane, with some participants suggesting methods to approach the problem. There is an indication that one participant has reached a conclusion about the coordinates of the closest point.

Contextual Notes

The original poster expresses uncertainty about how to proceed with the problem, and there is a lack of explicit equations or methods provided in the initial post.

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Homework Statement



Let P1 be the plane passing through the point A(5,-6,7) parallel to the xy-plane. Write down the coordinates of the point D on P1 which is closest to C(1,3,4).

Homework Equations



None.

The Attempt at a Solution



I am not sure how to attempt this problem. I tried sketching a diagram (attached), but didnt know what else to do.
 

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    Plane.jpg
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Have you tried deriving the equation of the plane P1 ?

The equation of a plane is given as follows
[tex]a(x-x_{0}) + b(y-y_{0}) +c(z-z_{0}) =0[/tex]
(a,b,c) is a normal vector to the plane.

It turns out that the distance between a point and a plane is shortest when the point is orthogonal to the plane.
 
It's a trick question. You know that the shortest distance from a point to a plane is in the normal direction to that plane. what is the normal to the plane they defined?
 
Thanks, I figured it out D(1,3,7) :)
 

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