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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
MCQ
If angle between the vectors $\vec{a}=2 \lambda^2 \hat{i}+4 \lambda \hat{j}+\hat{k}$ and $\vec{b}=7 \hat{i}-2 \hat{j}+\lambda \hat{k}$ is obtuse, then the values of $\lambda$ is
  • ✓
    $\left(0, \frac{1}{2}\right)$
  • B
    $\left(\frac{1}{2}, \infty\right)$
  • C
    $(-\infty, 0)$
  • D
    $\left[0, \frac{1}{2}\right]$
✓

Answer

Correct option: A.
$\left(0, \frac{1}{2}\right)$
Let $\theta$ be the angle between $\vec{a}$ and $\vec{b}$.
We know that, for obtuse angle $\theta, \cos \theta<0$
Also, $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta \quad$
i.e., $\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}<0$
$\Rightarrow \vec{a} \cdot \vec{b}<0 $
$\Rightarrow 14 \lambda^2-8 \lambda+\lambda<0$
$\Rightarrow 7 \lambda(2 \lambda-1)<0 $
$\Rightarrow 0<\lambda<\frac{1}{2}$
i.e. $\lambda \in\left(0, \frac{1}{2}\right)$

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