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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra2 Marks
Question
Given that $\vec{a} \cdot \vec{b}=0$ and $\vec{a} \times \vec{b}=\overrightarrow{0}$. What can you conclude about the vectors $\vec a $ and $\vec b$?
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Answer

Given that $\vec{a} \cdot \vec{b}=0$ 
$\Rightarrow|\vec{\mathrm{a}}| \cdot|\vec{\mathrm{b}}| \cdot \cos \theta=0$ (where $\theta$ is the angle between the vectors)
$\Rightarrow|\vec{\mathrm{a}}|=0$ or $|\vec{\mathrm{b}}|=0$ or $\cos \theta=0$ 
Also given that, $\vec{\mathrm{a}} \times \vec{\mathrm{b}}=0$ 
$\Rightarrow|\vec{\mathrm{a}}| \cdot|\vec{\mathrm{b}}| \cdot \sin \theta=0$ (where $\theta$ is the angle between the vectors)
$\Rightarrow|\vec{\mathrm{a}}|=0$ or $|\vec{\mathrm{b}}|=0$ or $\sin \theta=0$ 
As there is no value of $\theta$ for which both sin $\theta$ and cos $\theta$ are zero.
$\therefore$ The condition for which $\vec{a} \cdot \vec{b}=0$ and $\vec{\mathrm{a}} \times \vec{\mathrm{b}}=0$ is;
Either $|\vec{\mathrm{a}}|=0$ or $|\vec{\mathrm{b}}|=0$ 

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