Let a vector a=√2i-j+ k, 0, make an obtuse angle with the vector b=- ^2i+4√2j+4√2k and an…

Question

Let a vector $\overset{	o }{a}=\sqrt{2}\hat{i}-\hat{j}+\lambda \hat{k},\lambda >0$, make an obtuse angle with the vector $\overset{	o }{b}=-{\lambda }^{2}\hat{i}+4\sqrt{2}\hat{j}+4\sqrt{2}\hat{k}$ and an angle $	heta ,\frac{\pi }{6}<	heta <\frac{\pi }{2}$, with the positive $z$-axis. If the set of all possible values of $\lambda $ is $(\alpha ,\beta )-{\gamma }$, then $\alpha +\beta +\gamma $ is equal to $____$ .

Answer: 5

Accepted Answer

$\begin{gathered} {\vec{a}}^{\wedge }\hat{k}\in (\frac{\pi }{6},\frac{\pi }{2}) \ \frac{\lambda }{\sqrt{(2+1+{\lambda }^{2})}}\in (0,\frac{\sqrt{3}}{2})\Rightarrow 00\lambda \in (-3,3)\Rightarrow \lambda \in (0,3)\ldots \ldots \ldots .(1) \end{gathered}$

${\vec{a}}^{\wedge }\vec{b}$ is obtuse

$\begin{gathered} \vec{a}\cdot \vec{b}0\Rightarrow (\lambda -2{)}^{2}>0 \ \Rightarrow \lambda \in R-{2}\ldots \ldots .(2) \end{gathered}$

(1) \& (2) \Rightarrow  $\lambda \in (0,3)-{2}\Rightarrow \alpha +\beta +\gamma =5$

Solutions · 1