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Vector or Cross Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector or Cross Product5 Marks
Question
If $\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}},$ then verify that $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)=\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{a}}\times\vec{\text{c}}.$
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Answer

We have,
$\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}}$
$\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$
$\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}},$
$\big(\vec{\text{b}}+\vec{\text{c}}\big)=(\text{b}_1+\text{c}_1)\hat{\text{i}}+(\text{b}_2+\text{c}_2)\hat{\text{j}}+(\text{b}_3+\text{c}_3)\hat{\text{k}}$
Now, $\vec{\text{a}}\times\big(\vec{\text{b}}+\vec{\text{c}}\big)=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{a}_2&\text{a}_3\\\text{b}_1+\text{c}_{1}&\text{b}_2+\text{c}_2&\text{b}_3+\text{c}_3 \end{vmatrix}$
$=\hat{\text{i}}\big[\text{a}_2(\text{b}_3+\text{c}_3)-\text{a}_3(\text{b}_2+\text{c}_2)\big]-\hat{\text{j}}\big[\text{a}_1(\text{b}_3+\text{c}_3)-\text{a}_3(\text{b}_1+\text{c}_1)\big]\\+\hat{\text{k}\big[\text{a}}_1(\text{b}_2+\text{c}_2)-\text{a}_2(\text{b}_1+\text{c}_1)\big]$
$=\hat{\text{i}}\big[\text{a}_2\text{b}_3+\text{a}_2\text{c}_3-\text{a}_3\text{b}_2-\text{a}_3\text{c}_2\big]+\hat{\text{j}}\big[-\text{a}_1\text{b}_3-\text{a}_1\text{c}_3+\text{a}_3\text{b}_1+\text{a}_3\text{c}_1\big]\\+\hat{\text{k}}\big[\text{a}_1\text{b}_2+\text{a}_1\text{c}_2-\text{a}_2\text{b}_1-\text{a}_2\text{c}_1\big]\dots(1)$
$$$\vec{\text{a}}\times\vec{\text{b}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{a}_2&\text{a}_3\\\text{b}_1&\text{b}_2&\text{b}_3 \end{vmatrix}$
$=\hat{\text{i}}\big[\text{a}_2\text{b}_3-\text{a}_3\text{b}_2\big]+\hat{\text{j}}\big[\text{b}_1\text{a}_3-\text{a}_1\text{b}_3\big]+\hat{\text{k}}\big[\text{a}_1\text{b}_2-\text{a}_2\text{b}_1\big]\ \dots(2)$ 
$\vec{\text{a}}\times\vec{\text{c}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{a}_2&\text{a}_3\\\text{c}_1&\text{c}_2&\text{c}_3 \end{vmatrix}$
$=\hat{\text{i}}\big[\text{a}_2\text{c}_3-\text{a}_3\text{c}_2\big]+\hat{\text{j}}\big[\text{a}_3\text{c}_1-\text{a}_1\text{c}_3\big]+\hat{\text{k}}\big[\text{a}_1\text{c}_2-\text{a}_3\text{c}_1\big]\ \dots(3)$

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