Question

The value of $a$ for which the angle between $\overrightarrow{a}=2a^2\hat{i}+4a\hat{j}+\hat{k}$ and $\overrightarrow{b}=7\hat{i}-2\hat{j}+a\hat{k}$ is obtuse and the angle between $\overrightarrow{b}$ and $z$-axis is acute and less than $\frac{\pi }{6}$ is

• A. Does not exist
• B. Lies in $(0,\frac{1}{2})$
• C. Lies in $(-1,1)$
• D. Lies in $(0,1)$

Answer 1

The correct option is A Does not exist

Since the angle between $\overline{a}$ and $\overline{b}$ is obtuse, $\overline{a}.\overline{b}<0$.
$\therefore 14a^2-8a+a<0.$
$\therefore 14a^2-7a<0.$
$\therefore a(2a-1)<0,\Rightarrow 0<a<\frac{1}{2}$

Also, $\frac{(7\overline{i}-2\overline{j}+a\overline{k}).\left(\overline{k}\right)}{49+4+a^2}>\frac{\sqrt{3}}{2}$

$\Rightarrow 4a^2>159+3a^2$
$\Rightarrow a^2>159$
Thus we see that there is no intersection between the two conditions, or there is no value of a satisfying the question.