Question

the nine elements of the rotation matrix (r) are not all independent. write down the six dependencies between the elements.

Answer 1

There are six dependencies among the nine elements of a rotation matrix due to the requirements for orthogonality and orthonormality: three constraints from the columns being orthogonal unit vectors and three from the cross-column dot products being zero.

Step-by-step explanation:

The student is asking about the dependencies between the elements of a rotation matrix used in physics to represent the rotation of an object in three-dimensional space. A rotation matrix can be defined as:

R =
| r11 r12 r13 |
| r21 r22 r23 |
| r31 r32 r33 |

This matrix has nine elements, but not all are independent due to the properties of rotation matrices.

The six dependencies between the elements can be summarized as follows:

1. 
   
3. 
   
5. 
   
7. 
   

The rotation matrix must also be a proper rotation, meaning its determinant must be equal to 1. This is not a separate constraint, as it is a consequence of the orthogonality and orthonormality conditions.