I would be happy to get some help with the following problem:
let $a=(a_1, a_2,a_4,a_4)\in \Bbb R^4$,
and $M= \{x\in\Bbb R^4: |x|=1, \langle x, a \rangle =0\}, $
and $f(x_1 ,x_2, x_3, x_4) = \sum_{i=1}^4 a_i^2x_i$ .
I need to find the maximum of $f$ on the set $M$.
My Attempt:
I defined $g_1(x)=|x|^2-1$ and $g_2(x)=\langle x, a\rangle$.
Clearly $\nabla f(x)=(a_1^2, a_2^2, a_3^2,a_4^2), \\ \nabla g_1(x)=2x, \\ \nabla g_2(x)=a $
Now using lagrange multipliers i took $t, s\in\Bbb R$ that satisfy: $\nabla f(x) = t\cdot g_1(x) + s\cdot g_2(x)$
And by that I reached the following equstions: $ a_i^2 = 2t\cdot x_i + s\cdot a_i \ \ (\forall i=1,2,3,4)\\ x_1^2+x_2^2+x_3^2+x_4^2=1 \\ a_1\cdot x_1 + a_2\cdot x_2 + a_3\cdot x_3 + a_4\cdot x_4 = 0$
And that is where I stuck. From this point the algebra becomes complicated. This question is taken from a previous exam, and so I believe that the solution sould not be based on frustrating algebra...
Is my solution correct? (maybe I did some calculation errors on my attempt?) Is there an easier way to look at this problem? (any "tricks" I have missed?)
Thank you very much!