Let S be the set of a R for which the angle between u=a( b) i-6 j+3 k and v=( b) i+2 j+2a(…

Question

Let $S$ be the set of $a\in\mathbb R$ for which the angle between $\vec u=a(\ln b)\hat i-6\hat j+3\hat k$ and $\vec v=(\ln b)\hat i+2\hat j+2a(\ln b)\hat k$ ($b>1$) is acute. Then $S$ is:

Accepted Answer

Let $t=\ln b>0$ (since $b>1$). Acute angle requires $\vec u\cdot\vec v>0$:
$a t^2 - 12 + 6at > 0$

This must hold for arbitrary $b>1$, i.e. for all $t>0$. But as $t\to0^+$, the expression $\to -12<0$ regardless of $a$. So no value of $a$ works for all $t>0$.

$$S=\varnothing$$

Solutions · 1