# 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?