Question

Find the domain of $\lambda$ for which the vector $r=({\lambda }^2-4)i+2j+({\lambda }^2-9)k$ makes acute angles with the co-ordinate axes.

• A. $\lambda ϵ\{(-3,-1)\cup [0,3]\}$
• B. $\lambda ϵ\{(-3,0)\cup [1,3]\}$
• C. $\lambda ϵ\{(-3,-2)\cup [0,3]\}$
• D. $\lambda ϵ\{(-3,-2)\cup [2,3]\}$

Answer 1

The correct option is D $\lambda ϵ\{(-3,-2)\cup [2,3]\}$

Since $\overrightarrow{r}$ makes acute angle with $\hat{i},\hat{j},\hat{k}$ we have $\overrightarrow{r}.\hat{i}>0,\overrightarrow{r}.\hat{j}>0,$ and $\overrightarrow{r}.\hat{k}>0$
$\therefore {\lambda }^2-4>0,2>0,{\lambda }^2-9>0$
$\therefore (\lambda +2)(\lambda -2)>0$ and $(\lambda +3)(\lambda -3)>0$
$\therefore -2<\lambda$ or $\lambda >2$ and $-3<\lambda$ or $\lambda >3$
$\therefore \lambda ϵ(-3,-2)$ or $\lambda ϵ(2,3)$
$\therefore \lambda ϵ\{(-3,-2)\cup [2,3]\}$