![orthogonal matrix orthogonal matrix](https://slideplayer.com/slide/6065991/18/images/2/ORTHOGONAL+MATRICES+A+square+matrix+with+the+property+that+A−1+%3D+AT.jpg)
To represent a rotation and to translate to a rotated frame of reference.įor more information about how to use a matrix to represent rotations see When we think about the matrix in this way we can see how a matrix can be used So either orthogonal matrices or bivectors might be able to represent:Īll Orthogonal Matrices have determinants of 1 or -1 and all rotation matricesįor example |R| = cos(a) 2 + sin(a) 2 = 1 Rotations For a vector of dimension 'n' then the corresponding bivector will have dimension of n!/(n-2)! 2! This is related to bivectors in Geometric Algebra. So, using this formula, the degrees of freedom for a given dimension is: dimensions 'n' Which is the second entry in pascals triangle, or the number of combinations of 2 elements out of n. Which is an arithmetic progression as described on this page. So for an 'n' dimensional matrix the number of degrees of freedom is:
![orthogonal matrix orthogonal matrix](https://media.cheggcdn.com/media/a7b/a7b3c8cc-d6c9-4221-8c18-7408749f7d63/phpDBEFZ6.png)
When we add the third basis vector we have three more constraints: When we add the second basis vector we have two more constraints:
![orthogonal matrix orthogonal matrix](https://prod-qna-question-images.s3.amazonaws.com/qna-images/question/0e76e4ca-63bd-4702-9c48-a1a240d57366/80ffb753-5df9-4081-ab11-c25b68671744/3lyms5h_processed.jpeg)
So when we add the first basis vector we have one constraint: