The value of \( \lambda \) for which the angle between the vectors \( \vec{a}=2\lambda^{2}\hat{\imath}+4\lambda\hat{\jmath}+\hat{k} \) and \( \vec{b}=7\hat{\imath}-2\hat{\jmath}+\lambda\hat{k} \) is obtuse, is equal to :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \left(0,\displaystyle\frac{1}{2}\right) \)
The angle between two vectors is obtuse when their dot product is negative:
\( \displaystyle \vec{a}\cdot\vec{b}=14\lambda^{2}-8\lambda+\lambda \)
\( \displaystyle =7\lambda(2\lambda-1)\lt 0 \)
This holds for \( 0\lt\lambda\lt\frac{1}{2} \).
The angle can never be exactly \( 180^{\circ} \) here. That would need \( \vec{a}=k\vec{b} \) with \( k\lt 0 \); comparing the components gives \( k=-2\lambda \) and \( k\lambda=1 \), so \( -2\lambda^{2}=1 \), which has no real solution.
Hence \( \lambda\in\left(0,\frac{1}{2}\right) \).
Why the other options are incorrect
Option (1) \( \left[-\displaystyle\frac{1}{2},\displaystyle\frac{1}{2}\right] \) — Incorrect. It includes negative values of \( \lambda \), for which the dot product is positive.
Option (3) \( \left[0,\displaystyle\frac{1}{2}\right] \) — Incorrect. At \( \lambda=0 \) and \( \lambda=\frac{1}{2} \) the dot product is 0, which is a right angle.
Option (4) \( [0,1] \) — Incorrect. It includes values such as \( \lambda=1 \), for which the dot product is positive.