TITLE: Finding all scalars $k$ such that $\| kv \| = 10$
QUESTION [8 upvotes]: I have a homework question that asking to

Find all scalars $k$ such that $\|kv\| = 10$ when $v=(1,-4,6)$.

What I did that that I found the norm of $v$ which I found to be $\sqrt{53}$. Then I took that answer and multiplied by $k$ to get $10$ like this: $\sqrt{53}\cdot k=10$,
$$
k=\frac{10}{\sqrt{53}}
$$
I don't think this approach is right because it doesn't deal with the $k$ being calculated within the norm. I just don't know how to do it that was so any pointers would be greatly appreciated.

REPLY [4 votes]: If you don't already know $\| kv \| = |k| \cdot \| v \|$, you can first directly calculate $k\mathbf{v}$. This is just $k(1,-4,6)=(k,-4k,6k)$, so $\|k\mathbf{v}\|=\sqrt{k^2+16k^2+36k^2}$. Setting this equal to $10$, you find
$$
k^2=100/53,
$$
and then solve for possible values of $k$.