Q71 / 160 · AP EAMCET 2023 (15 May Shift 2) · Mathematics

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=|\vec{b}|=\sqrt{14}$ and $\vec{a} \cdot \vec{b}=-7$, then $\frac{|\vec{a} \times \vec{b}|}{|\vec{a} \cdot \vec{b}|}=$

  • A
    $7 \sqrt{3}$
  • Correct answer
    B
    $\sqrt{3}$
  • C
    $49 \sqrt{3}$
  • D
    $\frac{\sqrt{3}}{7}$

Solution

Given, $\vec{a} \cdot \vec{b}=-7 \Rightarrow|\vec{a}||\vec{b}| \cos \theta=7$ $\Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \tan \theta=\sqrt{3}$ Now, $\frac{|\vec{a} \times \vec{b}|}{|\vec{a} \cdot \vec{b}|}=\frac{|\vec{a}||\vec{b}| \sin \theta}{|\vec{a}||\vec{b}| \cos \theta}=\tan \theta=\sqrt{3}$

Hence, the correct answer is option: 'B'.

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