Can Matrix Determinants Be Used to Find Optimal Area in Higher Dimensions?

In summary, the conversation discusses the possibility of finding the area of a triangle or parallelogram in euclidean space using a matrix determinant composed of unity, x coefficients, and y coefficients. It is also mentioned that this method may not be as simple in higher dimensions and that using a 3D matrix would give volume instead of area. The conversation then shifts to a problem involving a rectangular table and a triangular sheet, discussing the optimal angle and position of the triangle to maximize the area on the table.
  • #1
vin300
603
4
It is possible to find area of triangle or parallelogram in euclidean by using matrix determinant composed of unity, x coeffs and y coeffs in row1,2,3 respectively. Is it possible to do that in higher dimensions as well although it may be not as simple as in 2D case. In 3d matrix composed of x,y,z would instead give volume but I am looking for area of a plane suspended in space.
 
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  • #2
vin300 said:
It is possible to find area of triangle or parallelogram in euclidean by using matrix determinant composed of unity, x coeffs and y coeffs in row1,2,3 respectively. Is it possible to do that in higher dimensions as well although it may be not as simple as in 2D case. In 3d matrix composed of x,y,z would instead give volume but I am looking for area of a plane suspended in space.
It's all explained here:

http://math.arizona.edu/~calc/Text/Section13.4.pdf
 
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  • #3
I am thinking about a newer more interesting and seemingly difficult problem. The problem has a rectangular table with a triangular sheet to be laid upon, whose dimensions are greater than the table but the area is lesser. What would be the relative angle, and position of the triangle that gives optimum area on the field, minimum outside?
 

1. How do you find the area of a matrix?

The area of a matrix can be found by multiplying the number of rows by the number of columns. For example, a 3x5 matrix has an area of 15 units.

2. Can the area of a matrix be negative?

No, the area of a matrix is always a positive value. This is because it represents the number of elements within the matrix, which cannot be negative.

3. What is the difference between finding area with a matrix and finding area with regular shapes?

When finding area with a matrix, we are counting the number of elements within the matrix. With regular shapes, we use formulas to calculate the area based on the dimensions of the shape.

4. Can the area of a matrix be a decimal or fraction?

Yes, the area of a matrix can be a decimal or fraction if the dimensions of the matrix are not whole numbers. For example, a 2.5x3 matrix has an area of 7.5 units.

5. How is the area of a matrix used in real life?

The concept of finding area with a matrix is used in various fields such as computer science, engineering, and economics. It is used to represent and analyze data in a structured way, making it easier to make calculations and predictions.

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