The matrix A is:
A=[]Since A is an orthogonal matrix, we know that:
AT=A−1⇒ATA=IThis property tells us that
A is orthogonal, and it implies that
ATA (the product of
A′ s transpose and
A is equal to the identity matrix
I, which is:
ATA=[]Now, let's calculate
ATA step by step. The transpose of matrix
A, denoted
AT is:
AT=[] Now, we perform matrix multiplication between
AT and A :
ATA=[][]Performing this multiplication, we get the following matrix:
ATA=[| y2+z2+x2 | xy+zx+xy | xz+yz+x2 |
| xy+zx+xy | x2+x2+y2 | xy+xz+yz |
| xz+yz+x2 | xy+xz+yz | x2+y2+z2 |
]This matrix must be equal to the identity matrix I, which is:
[]By comparing the elements of the matrices, we get the following system of equations:
1. y2+z2+x2=12⋅xy+zx+xy=03⋅xz+yz+x2=1Thus, the key result from the orthogonality condition is:
x2+y2+z2=1