What inference can you draw if a b =0 and a . b =0. - Vidyadip

Question

What inference can you draw if $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$ and $\vec{\text{a}}.\vec{\text{b}}=0.$

✓

Answer

Given:
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\vec{0}$
$\Rightarrow\vec{\text{a}}=0$
$\vec{\text{b}}=0$
$\therefore\vec{\text{a}}||\vec{\text{b}}$
Also,
$\vec{\text{a}}.\vec{\text{b}}=0$
$\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$
$\Rightarrow\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0}$ or, $\vec{\text{a}}\perp\vec{\text{b}}$
But $\vec{\text{a}}$ cannot be both perpendicular as well as parallel to $\vec{\text{b}}.$
$\therefore|\vec{\text{a}}|=0$
$\big|\vec{\text{b}}\big|=0$

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