If $|\bar{a}|=\sqrt{26},|\bar{b}|=7$ and $|\bar{a} \times \bar{b}|=35$, then $\bar{a} \cdot \bar{b}=$

If $|\bar{a}|=\sqrt{26},|\bar{b}|=7$ and $|\bar{a} \times \bar{b}|=35$, then $\bar{a} \cdot \bar{b}=$
  1. $\pm 5 \sqrt{26}$
  2. $\pm \sqrt{26}$
  3. $\pm 7$
  4. $\pm 7 \sqrt{26}$

Solution

$\begin{aligned} & |\vec{a}|=\sqrt{26},|\vec{b}|=7,|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta=35 \\ & \Rightarrow \sqrt{26} \times 7 \sin \theta=35 \\ & \Rightarrow \sin \theta=\frac{5}{\sqrt{26}} \\ & \Rightarrow \cos \theta= \pm \frac{1}{\sqrt{26}}\end{aligned}$ Now $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta=\sqrt{26} \times 7 \times \frac{ \pm 1}{\sqrt{26}}= \pm 7$

Asked in: MHT CET 2022 (07 Aug Shift 2)

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