Practicing Success
If vectors $ax \hat{i}+3 \hat{j}-5 \hat{k}$ and $x \hat{i}+2 \hat{j}+2 a x \hat{k}$ make an acute angle with each other, for all x ∈ R, then a belongs to the interval |
$\left(-\frac{1}{4}, 0\right)$ $(0,1)$ $\left(0, \frac{6}{25}\right)$ $\left(-\frac{3}{25}, 0\right)$ |
$\left(0, \frac{6}{25}\right)$ |
Since vectors make an acute angle with each other so their dot product must be positive i.e. $a x^2-10 a x+6>0 ~\forall~ x \in R$ $\Rightarrow-ax^2+10 ax-6<0 ~\forall~ x \in R \Rightarrow-a<0$ and $100 a^2<24 a$ Hence (3) is correct answer. |