# Finding all scalars $k$ such that $\| kv \| = 10$

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.

By the definition of the norm you have $$10=\|kv\| = |k|\cdot\|v\| = |k|\sqrt{53}$$ so $|k| = \frac{10}{\sqrt{53}}$ and $k_1 = -\frac{10}{\sqrt{53}}$, $k_2 = \frac{10}{\sqrt{53}}$.
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 $k$.