Given that the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ (no two of them are collinear). Further if $

Question

Given that the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ (no two of them are collinear). Further if $(\vec{a}+\vec{b})$ is collinear with $\vec{c},(\vec{b}+\bar{c})$ is collinear with $\vec{a}$ and $|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{2}$. Then the value of $|\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}|$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

$\vec{a}+\vec{b}=\lambda \vec{c}$ and $\vec{b}+\vec{c}=\mu \vec{a}$
$(\vec{a}-\vec{c})=\lambda \vec{c}-\mu \vec{a} \Rightarrow \vec{a}(1+\mu)=\vec{c}(\lambda+1) \Rightarrow \mu=1, \lambda=-1 \Rightarrow \vec{a}+\vec{b}+\vec{c}=0$
$|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0 ; \quad \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-3$