Question

http://i.imgur.com/9ogElpu.png
Help!

Answer 1

The set of equations forms an indeterminate system. So, first, simplify,
\[x+z=6\\
x+y+2z=12\Rightarrow\\
x+z=6\\
y+z=6\]
Let z be a parameter, for example, lambda,
\[z=\lambda\Rightarrow \vec{v}=(x,y,z)=(\lambda,\lambda,6-\lambda)\]
Then calculate the modulus,
\[|\vec{v}|=\sqrt{3(\lambda^2-4\lambda+12)}\]
Then calculate the derivative and find its zeros, to find the maximum,
\[\frac{d\vec{v}}{d\lambda}=\frac{3\lambda-6}{\sqrt{3(\lambda^2-4\lambda+12)}}\Rightarrow \lambda=2\]
So,
\[|\vec{v}|=2\sqrt{6}\]

Answer 2

Thank you. :)