If $\vec{a}=x \hat{i}+(x-1) \hat{j}+\hat{k}$ and $\vec{b}=(x+1) \hat{i}+\hat{j}+a \hat{k}$ always make an acute angle with each other for every value of $x \in R$, then:
Answer & explanation
Correct answer: option 2
$\vec{a} . \vec{b}=(x \hat{i}+(x-1) \hat{j}+\hat{k}) . ((x+1) \hat{i}+\hat{j}+a \hat{k})$
$=x(x+1)+x-1+a$
$=x^2+2 x+a-1$
We must have
$\vec{a} . \vec{b}>0 ~\forall~ x \in R$
$x^2+2 x+a-1>0$
$(x+1)^2+(a-2)>0$
min. value = 0
so $a -2 > 2$ always $⇒a>2$
Hence (2) is correct answer.