Transformation of Matrix onto plane

In summary, the matrix for the transformation that projects each point in R3 perpendicularly onto the plane 7x + y + 3z = 0 is obtained by using the normal vector of the plane, n, as the first column of the matrix, and the projection vector, v', as the second column. The equation n <dot> v = 0 represents the equation of the plane, and v represents all the possible points in R3 that are projected onto the normal vector. The v-projection is used to obtain the projection of v onto the plane, and v' represents the solution because it is the vector that lies on the plane and is closest to v.
  • #1
FlorenceC
24
0
Find the matrix for the transformation that projects each point in R3 (3-D) perpendicularly onto the plane 7x + y + 3z = 0 .
The attempt at a solution is attached for question 1 (actually instructor's solution)

I kind of understand it but ...
why is n <dot> v = equation of the plane?
Does v represent all of the possible points of R^3 (certainly does not seem so...) which is projected to the normal?
I understand the v-projection is there to get the projection of v onto the plane because we cannot directly project to the plane right? But why do we want v' what does v' represent and how is that the solution?
 

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  • #2
I moved your first post to the Linear Algebra subsection of the technical math forum. We ask that members not post photos of their work, but instead post the work itself in the input pane. I might have let it slide, but the document you posted is eleven handwritten pages long, which is unreasonably long. Some helpers will not even bother looking at work in attached files.

I have locked this thread - please ask focused questions in the other thread, which is here: https://www.physicsforums.com/threads/linear-algebra-matrix-transformation-to-plane.778451/
 

1. What is a transformation of matrix onto plane?

A transformation of matrix onto plane is a mathematical process that involves mapping points from a matrix onto a two-dimensional plane. This allows for a visual representation of the data in the matrix and can help in understanding patterns and relationships within the data.

2. How is a transformation of matrix onto plane performed?

A transformation of matrix onto plane is performed by multiplying each point in the matrix by a transformation matrix, which is a set of numbers that dictate how the points will be mapped onto the plane. The resulting points will then be plotted on the plane, creating a visual representation of the original matrix.

3. What are some common types of transformations used in matrix onto plane?

Some common types of transformations used in matrix onto plane include translation, rotation, scaling, and shearing. Translation involves moving the points in a specific direction, while rotation involves rotating the points around a fixed point. Scaling involves changing the size of the points, and shearing involves skewing the points in a certain direction.

4. What is the purpose of a transformation of matrix onto plane?

The purpose of a transformation of matrix onto plane is to visualize and analyze data in a matrix in a more intuitive and understandable way. It allows for the identification of patterns and relationships within the data and can aid in making predictions or drawing conclusions from the data.

5. How is a transformation of matrix onto plane used in real-world applications?

A transformation of matrix onto plane has various real-world applications, such as computer graphics, computer vision, and data analysis. It is commonly used in creating 3D computer models, image processing, and analyzing data in fields such as economics, engineering, and biology.

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