If a . a =0 and a . b =0, what can you conclude about the vector … - Vidyadip

Question

If $\vec{\text{a}}.\vec{\text{a}}=0$ and $\vec{\text{a}}.\vec{\text{b}}=0,$ what can you conclude about the vector $\vec{\text{b}}?$

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Answer

Given that $\vec{\text{a}}.\vec{\text{a}}=0$
$\Rightarrow|\vec{\text{a}}|^2=0$
$\Rightarrow|\vec{\text{a}}|=0\dots(1)$
Also, given that
$\vec{\text{a}}.\vec{\text{b}}=0$
$\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$ (where $\theta$ is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$)
$\Rightarrow0\big|\vec{\text{b}}\big|\cos\theta=0$ [From (1)]
$\Rightarrow0=0$
So, it means that for any vector $\vec{\text{b}},$ the given equation $\vec{\text{a}}.\vec{\text{b}}=0$ is satisfid.

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