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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.

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(+1) for showing work. – The Chaz 2.0 Nov 1 '11 at 18:52
up vote 8 down vote accepted

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

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So I was right! Thank you this makes more sense now that I see it – Cheesegraterr Nov 1 '11 at 19:03
@Cheesegraterr: yes, you were right - just lost one solution. You're welcome – Ilya Nov 1 '11 at 19:17

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 $k$.

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