This is a simple problem I am having a bit of trouble with. I am not sure where this leads.
Given that $\vec a = \begin{pmatrix}4\\-3\end{pmatrix}$ and $|\vec b|$ = 3, determine the limits between which $|\vec a + \vec b|$ must lie.
Let, $\vec b = \begin{pmatrix}\lambda\\\mu\end{pmatrix}$, such that $\lambda^2 + \mu^2 = 9$
Then,
$ \begin{align} \vec a + \vec b &= \begin{pmatrix}4+\lambda\\-3 + \mu\end{pmatrix}\\ |\vec a + \vec b| &= \sqrt{(4+\lambda)^2 + (\mu - 3)^2}\\ &= \sqrt{\lambda^2 + \mu^2 + 8\lambda - 6\mu + 25}\\ &= \sqrt{8\lambda - 6\mu + 34} \end{align} $
Then I assumed $8\lambda - 6\mu + 34 \ge 0$. This is as far I have gotten. I tried solving the inequality, but it doesn't have any real roots? Can you guys give me a hint? Thanks.