Question

Among all the unit vectors u= [x y z] find the one for which the sum of x+2y+5z is minimal

Answer 1

um... how about \(\vec e=\langle 1,0,0\rangle\)
(not sure I understand the question)

Answer 2

This will do it
\[\left(-\frac{1}{\sqrt{30}},-\sqrt{\frac{2}{15}},-\sqrt{\frac{5}{6}}\right)
\]
The minimum of x+2y+5z will be about -5.47723.

One is tempted to take (0,0,-1), but it is not right.
It is easy to work with spherical coordinates. All unit vectors look like
\[(\cos (\theta ) \sin (\phi
),\sin (\theta ) \sin (\phi
),\cos (\phi )),\quad 0\le \theta \le 2\pi, 0\le \phi \le \pi
\]
try to minimize the quantity\[
x+2y+5z =2 \sin (\theta ) \sin (\phi
)+\cos (\theta ) \sin (\phi
)+5 \cos (\phi )

\]
I leave the details for you.

Answer 3

I would just use Lagrange multipliers on the original equations/function

Answer 4

You can also do that. Probably do it and show the asker the details.

Answer 5

thank you
ill work on getting the details