If a =0 and a =0, then what can be concluded about the vector b? - Vidyadip

Question

$\text{If}\ \vec{a}\cdot\vec{a}=0\ \text{and}\ \vec{a}\cdot\vec{b}=0,$ then what can be concluded about the vector $\vec{b}?$

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Answer

$\text{Given:}\ \ \vec{a}.\vec{a}=0\Rightarrow\ \ \big|\vec{a}\big|^2=0$ $ \ \Rightarrow\ \ \ \ \big|\vec{a}\big|=0$ $\text{Again}\ \ \vec{a}.\vec{b}=0\ \Rightarrow\ \ \big|\vec{a}\big|.\Big|\vec{b}\Big|\text{cos}\ \theta=0$ $\ \Rightarrow\ \ 0.\Big|\vec{b}\Big|\text{cos}\theta=0\ \Big[\because\big|\vec{a}\big|=0\Big]$ $\Rightarrow\ $ 0 = 0 for all (any vector $\vec{b}.$)Therefore, $\vec{b}$ can be any vector.

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