KMO 2001 Problem 3

Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$ . Prove that $$(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|$$ and find all cases of equality.