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Vector or Cross Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product2 Marks
Question
For any three vectors $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ write the value of $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)+\vec{\text{b}}\times\big(\vec{\text{c}}+\vec{\text{a}}\big)+\vec{\text{c}}\times\big(\vec{\text{a}}+\vec{\text{b}}\big).$
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Answer

$\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)+\vec{\text{b}}\times\big(\vec{\text{c}}+\vec{\text{a}}\big)+\vec{\text{c}}\times\big(\vec{\text{a}}+\vec{\text{b}}\big)$
$=\big(​\vec{\text{a}}\times​\vec{\text{b}}\big)+\big(​\vec{\text{a}}\times​\vec{\text{c}}\big)+\big(​\vec{\text{b}}\times​\vec{\text{c}}\big)+\big(​\vec{\text{b}}\times​\vec{\text{a}}\big)+\big(​\vec{\text{c}}\times​\vec{\text{a}}\big)+\big(​\vec{\text{c}}\times​\vec{\text{b}}\big)$
$=\big(​\vec{\text{a}}\times​\vec{\text{b}}\big)+\big(​\vec{\text{a}}\times​\vec{\text{c}}\big)+\big(​\vec{\text{b}}\times​\vec{\text{c}}\big)-\big(​\vec{\text{a}}\times​\vec{\text{b}}\big)-\big(​\vec{\text{a}}\times​\vec{\text{c}}\big)-\big(​\vec{\text{b}}\times​\vec{\text{c}}\big)$
$=\vec{0}$

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