Let $\vec{b}=3 \hat{i}-2 \hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be two vectors. If $\vec{a}$…

Let $\vec{b}=3 \hat{i}-2 \hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be two vectors. If $\vec{a}$ is a vector such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, then $|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|=$
  1. $15$
  2. $\sqrt{261}$
  3. $\sqrt{234}$
  4. $33$

Solution

$\begin{aligned} & \text {} \because \vec{a}+\vec{b}+\vec{c}=0 \Rightarrow \vec{a}=-(\vec{b}+\vec{c}) \\ & \text { Now, } \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=-(\vec{b}+\vec{c}) \times \vec{b}+\vec{b} \times \vec{c}-\vec{c} \times(\vec{b}+\vec{c}) \\ & =-(\vec{b} \times \vec{b}+\vec{c} \times \vec{b})+\vec{b} \times \vec{c}-\vec{c} \times \vec{b}-\vec{c} \times \vec{c}=3(\vec{b} \times \vec{c})\end{aligned}$ $\begin{aligned} & \text { Now, }(\vec{b} \times \vec{c})=(3 \hat{i}-2 \hat{j}+\hat{k}) \times(\hat{i}-\hat{j}-\hat{k})=3 \hat{i}+4 \hat{j}-\hat{k} \\ & \therefore \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=3(3 \hat{i}+4 \hat{j}-\hat{k})=9 \hat{i}+12 \hat{j}-3 \hat{k} \\ & |\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|=\sqrt{81+144+9}=\sqrt{234}\end{aligned}$

Asked in: MHT CET Full Test 13

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