Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
$\vec{a}$ and $\vec{b}$ are two unit vectors inclined at an angle $\alpha(\alpha \in [0, \pi])$ to each other and $|\vec{a} + \vec{b}| < 1$ then $\alpha$ can lie in:
\alpha \in \left[\frac{\pi}{3}, \frac{2\pi}{3}\right]
\alpha \in \left[\frac{\pi}{3}, \frac{\pi}{2}\right]
\alpha \in \left[\frac{2\pi}{3}, \frac{5\pi}{6}\right]
\alpha \in \left[\frac{2\pi}{3}, \frac{5\pi}{7}\right]
Step-by-Step Solution
Key Concept: Linear independence is verified via non-zero determinant; dot product sign determines angle type; coplanarity corresponds to zero determinant.
Given $|\vec{a}+\vec{b}|0$ with $\vec{b}$ making obtuse angle, implying $\lambda<0$. Condition (C) requires solving the determinant $\begin{vmatrix}2 & -1 & 1 \\ 1 & 2 & 1+a \\ 3 & a & 5\end{vmatrix}=0$.
Correct Answer: 1,2