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  4. If veca+2 vecb+3 vecc = vec0, then veca × vecb+ vecb × vecc+ vecc × veca=

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Q. If $\vec{a}+2\vec{b}+3\vec{c} = \vec{0}$, then $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=$

Solution:

$\vec{a}+2 \vec{b}+3 \vec{c}=0$
$\Rightarrow \vec{a}=-2 \vec{b}-3 \vec{c}$
$\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}$
$=(-2 \vec{b}-3 \vec{c}) \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times(-2 \vec{b}-3 \vec{c})$
$=-2(\vec{b} \times \vec{b})-3(\vec{c} \times \vec{b})+\vec{b} \times \vec{c}-2(\vec{c} \times \vec{b})-3(\vec{c} \times \vec{c})$
$=-0+3(\vec{b} \times \vec{c})+\vec{b} \times \vec{c}+2(\vec{b} \times \vec{c})+0$
$=6(\vec{b} \times \vec{c})$