I am having difficulties to solve the following question. Would you please help me?

suppose that X and Y are independent and each rotationally invariant on R^k

a) Let p denote any orthogonal projection with dim P = k1

determine the distribution of the correlation coefficient r= X'PY/(|PX||PY|)

I think r is a special case of ∑(Xi-barX)(Yi-barY) = X'PX where P = I-n^(-1)11'

but what should I do after?

b) Let X = X1 with Xi on R^k1 where i =1,2 and prove

X2

- Each Xi is rotationally invariant in its own right

- If Vi is in R[from K1 to K] with V'V = I, then V'X = X1

I tried so hard to solve those two but i got stuck. Please help me