Question

If ||u||= 4 , ||v||=3 and u*v = 3, find ||u+2v||

Answer 1

expand ||u + 2u||

Answer 2

For this, you'll need to use a particular property of the norm of a vector. Specifically, if \(\vec{w}\) is a vector, then \[||\vec{w}||=\sqrt{w\cdot w}\]Replace \(\vec{w}\) with \(u+2v\), and suddenly everything will fall out.

Answer 3

One more thing you might want to remember, is that if you take \((u+2v)\cdot(u+2v)\), you can expand that using FOIL.

Answer 4

I worked it out so that I had \[u*u + (2v*u)(2v*u)\] but I feel like that's going in the completely wrong direction. Or is that what I want to end up getting?
I worked it through and got \[\sqrt{96}\] as an answer

Answer 5

That doesn't look right. Let's go through this step by step.\[\begin{aligned}
||u+2v||&=\sqrt{(u+2v)\cdot(u+2v)}\\
&=\sqrt{u\cdot u+4u\cdot v+4v\cdot v}\\
&=\sqrt{||u||^2+4(u\cdot v)+4||v||^2}.
\end{aligned}\]Now that we've reached this point, we just have to plug in numbers.\[\begin{aligned}
||u+2v||&=\sqrt{16+4\cdot3+9}\\
&=\sqrt{16+12+9}\\
&=\sqrt{37}.
\end{aligned}\]Did you follow all of these steps?

Answer 6

Yes, I see it now! Thanks so much!