Check my work regarding vector magnitude and direction?

Sorry if this is the wrong section, but it came from my Calc book.

The question is : find a vector of norm $\displaystyle 10\sqrt{2}$ in the direction opposite of $\displaystyle 3i + 4j - 5k$.

As I understand it, the direction vector is the $\displaystyle \mathbf{V}/\left \| \mathbf{V} \right \|$

So the direction vector of the given vector is:

$\displaystyle 3i/\sqrt{50} + 4j/\sqrt{50} - 5k/\sqrt{50}$ right?

New vector needs to be opposite so switch the signs:

$\displaystyle -3i/\sqrt{50} - 4j/\sqrt{50} + 5k/\sqrt{50}$

Now do I multiply the required norm into this direction vector to get the final answer:?

$\displaystyle -30\sqrt{2}i/\sqrt{50} - 40\sqrt{2}j/\sqrt{50} + 50\sqrt{2}k/\sqrt{50}$

I just took a stab at it, so I likely messed up...