Orthogonal Transformation and condition

[eoghan]

Hi there!
In order to proof the orthogonal condition a_ija_ik=$$\delta_{jk}$$ j,k=1,2,3
I write the invariance of the length of a vector in two coordinate systems:
x'_ix'_i=x_ix_i
Using the linear transformation:
x'_i=a_i1x_i1+a_i2x_i2+a_i3x_i3
the first term becomes:
a_ija_ikx_jx_k

My question is: why can't I write
a_ij²=x_j²

 

[pat_connell]

xk2to prove the orthogonal condition?The answer is that you cannot write aij2=xj2xk2 to prove the orthogonal condition because this equation does not necessarily demonstrate the equality of the two sides of the equation. The orthogonal condition requires that aijaik is equal to the Kronecker delta, δjk. To demonstrate the orthogonal condition, you must use the linear transformation and the invariance of the length of a vector to show that aijaik is equal to δjk.