multivariate calculation – Why does the Lagrange multiplier method fail?

Question

Let $ mathbf {x} $ and $ mathbf {y} $ to be two vectors in $ mathbb {R} ^ 2 $, and $ f $ to be a function defined by $ f ( mathbf {x}, mathbf {y}): = mathbf {x} cdot mathbf {y} $. Can this function be minimized subject to the conditions $ parallel mathbf {x} parallel = parallel mathbf {y} parallel = $ 1, with the Lagrange multiplier method?

Attempt

I have failed to use the Lagrange Multiplier to solve this issue. The main reason is that as I try to express $ f $ according to four variables $ x_1, x_2, x_3, x_4 $, the gradient of $ g_1 $ and the gradient of $ g_2 $ has the last two entries being $ 0. While the gradient of $ f $ does not have zero element. So, it seems that there can be no $ lambda_1 $ and $ lambda_2 $ this is equivalent to the three gradient vectors. Am I right? Or am I grossly simplifying the question when I express $ f $ according to four variables?