Vector Algebra
Acute Angle Between Vectors
nta_pyq_2024_jan
Grade 12
Question:
The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat{i}-2\hat{j}+2\hat{k}$ and $\alpha\hat{i}+2\alpha\hat{j}-2\hat{k}$ is acute, is ___.
Step-by-Step Solution
Key Concept: Angle is acute when the dot product $> 0$: $(\alpha\hat{i}-2\hat{j}+2\hat{k})\cdot(\alpha\hat{i}+2\alpha\hat{j}-2\hat{k})=\alpha^2-4\alpha-4>0$.
$\alpha^2-4\alpha-4>0\Rightarrow(\alpha-2)^2>8$. For positive $\alpha$: $\alpha>2+2\sqrt{2}\approx4.83$. Least positive integer $\alpha=5$.
Correct Answer: 5