Given three vectors $\bar{a}, \bar{b}, \bar{c}$, two of which are collinear. If $\bar{a}+\bar{b}$ is…

Given three vectors $\bar{a}, \bar{b}, \bar{c}$, two of which are collinear. If $\bar{a}+\bar{b}$ is collinear with $\bar{c}$ and $\bar{b}+\bar{c}$ is collinear with $|\bar{a}|=|\bar{b}|=|\bar{c}|=\sqrt{2}$, then $\bar{a} \cdot \bar{b}+\bar{b} \cdot \bar{c}+\bar{c} \cdot \bar{a}=$
  1. $5$
  2. $-3$
  3. $3$
  4. $-1$

Solution

$\begin{aligned} & \vec{a}+\vec{b}=\lambda \vec{c} \quad \ldots(i) \text { and } \vec{b}+\vec{c}=\mu \vec{a} \quad \ldots(i i) \\ & \Rightarrow \vec{a}-\vec{c}=\lambda \vec{c}-\mu \vec{a} \quad \text { form (i) and (ii) } \\ & \Rightarrow(1+\mu) \vec{a}=(1+\lambda) \vec{c} \\ & \Rightarrow \mu=\lambda=-1 \\ & \Rightarrow \vec{a}+\vec{b}+\vec{c}=0 \\ & \Rightarrow|\vec{a}+\vec{b}+\vec{c}|^2=|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}) \\ & \Rightarrow 0=2+2+2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}) \\ & \Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-3\end{aligned}$

Asked in: MHT CET 2022 (07 Aug Shift 1)

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