Vector Algebra
Obtuse Angle — Range of α
nta_pyq_2024_apr
Grade 12
Question:
The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t\in\mathbb{R}$, is
$\left(-\dfrac{4}{3},1\right)$
$[0,1)$
$\left(-\dfrac{4}{3},0\right]$
$(-2,0]$
Step-by-Step Solution
Key Concept: $\vec{a}\cdot\vec{b}<0$ for all $t\in\mathbb{R}$: $\alpha t^2-12-6\alpha t<0\,\forall\,t$. Need $\alpha<0$ and discriminant $<0$: $36\alpha^2+48\alpha<0\Rightarrow12\alpha(3\alpha+4)<0\Rightarrow-4/3<\alpha<0$.
$\alpha\in(-4/3,0]$.
Correct Answer: 3