Show that (a-b) (a+b)=2(a b ) - Vidyadip

Question

Show that
$(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=2(\vec{a}\times{\vec{b}})$

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Answer

$\text{L.H.S}=(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=\vec{a}\times\vec{a}+\vec{a}\times\vec{b}-\vec{b}\times\vec{a}-\vec{b}\times\vec{b}$ $\ \Big[\because\vec{a}\times\vec{a}=\vec{0},\vec{b}\times\vec{b}=\vec{0},\vec{b}\times\vec{a}=-\vec{a}\times\vec{b}\Big]$
$=\vec{0}+\vec{a}\times\vec{b}+\vec{a}\times\vec{b}-\vec{0}=2\vec{a}\times\vec{b}=\text{R.H.S.}$

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