Length Vector Rotation Matrix Constraints | Preserve A

[mind game]

what constraints must the elements of three dimensional rotation matrix satisfy in order to preserve length of vector A

 

[robphy]

Here's a start:
your question expressed symbolically...
$${\vec A'}^\top\vec A'=(R\vec A)^\top(R\vec A)=\vec A^\top\vec A$$

So, what must $$R$$ satisfy? Can you do matrix algebra?

 

[quicknote]

?

The elements of a three-dimensional rotation matrix must satisfy the following constraints in order to preserve the length of vector A:

1. Orthogonality: The columns (or rows) of the matrix must form an orthogonal basis. This means that each column (or row) must be perpendicular to the other columns (or rows) in the matrix.

2. Unit length: Each column (or row) must have a length of 1. This ensures that the matrix does not scale the vector A.

3. Determinant of 1: The determinant of the matrix must be 1. This ensures that the matrix does not reflect or invert the vector A.

By satisfying these constraints, the rotation matrix will preserve the length of vector A, meaning that the magnitude of vector A will remain unchanged after it is rotated by the matrix. This is important in scientific and mathematical calculations that involve rotating vectors, as it ensures the accuracy and validity of the results.