Vector calculus: Projection of a point to a plane

In summary: So, in summary, the problem involves finding the projection of a point onto a plane defined by a point and a normal vector. There are multiple methods to solve this problem, including finding the projection vector and subtracting it from the original point, or solving for the distance from the point to the plane and adding a vector in the same direction as the normal vector with that distance.
  • #1
willworkforfood
54
0
The problem reads as follows:

"The projection of a point P = (x,y,z) to a plane is a point on the plane that is closest to P. If the plane is defined by a point P0 = (x0,y0,z0) and a normal vector n=(x1,y1,z1), computer the projection of P on this plane."

Well, I haven't had a relevant Calculus course in many years, but I'm 99.9% certain that this is a vector calculus problem. My memory is a little sketchy on how to solve for a projection of a point on to a plane, so could anyone here perhaps provide a forumla, algorithm, solution or some other explanation of this problem? Thank you all very much for your time and help even if you don't reply! :)
 
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  • #2
First find the projection Q of P-P0 onto n. If you think about it geometrically this is the vector connecting P and the closest point on the plane. Then if you take P - Q, you should be able to see that this is the projection of P onto the plane.
 
  • #3
its not a vector calculus problem its a 3D mathematics problem or a lin alg problem. You could do either orthodontist method which works good or you can try one of two other methods
[0] you can find teh equation that passes between the point P and projP. Which is easy then substitue back into the plane equation and solve for t.
lot mor work but its worth it
[1] (this is just another way to reword orthodontist, but i think more in lamens terms)
solve for the distance D of P to the plane. which is easy then add a "vector" with length with that distance
 

1. What is a vector in vector calculus?

A vector in vector calculus is a mathematical object that has both magnitude (size) and direction. It is represented by an arrow in three-dimensional space and can be used to represent physical quantities such as displacement, velocity, and force.

2. What is a plane in vector calculus?

A plane in vector calculus is a two-dimensional surface that extends infinitely in all directions. It can be defined by a point and two non-collinear vectors, known as the normal vectors, which are perpendicular to the plane and determine its orientation.

3. How do you project a point onto a plane in vector calculus?

To project a point onto a plane in vector calculus, you first need to find the normal vectors of the plane. Then, you can use the dot product to calculate the distance from the point to the plane, and finally, use this distance to find the projected point on the plane.

4. What is the purpose of projecting a point onto a plane in vector calculus?

The purpose of projecting a point onto a plane in vector calculus is to find the closest point on the plane to the given point. This can be useful in various applications, such as computer graphics, engineering, and physics.

5. Can a point be projected onto any plane in vector calculus?

Yes, a point can be projected onto any plane in vector calculus as long as the plane is defined by a point and two non-collinear vectors. However, the resulting projected point may be outside the boundaries of the plane if the plane is not infinite.

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