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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA2 Marks
Question
Let the vectors $\vec{a},\vec{b},\vec{c}$ be given as $a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k},\ b_{1}\hat{i}+b_{2}\hat{j}+b_{3}\hat{k},$ $c_{1}\hat{i}+c_{2}\hat{j}+c_{3}\hat{k}.$ Then show that  $\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}.$
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Answer

Given:  $\text{Vector}\ \vec{a}=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k},\ \vec{b}=b_{1}\hat{i}+b_{2}\hat{j}+b_{3}\hat{k}$ $\text{and}\ \vec{c}=c_{1}\hat{i}+c_{2}\hat{j}+c_{3}\hat{k}$

$\therefore\ \ \vec{b}+\vec{c}=(b_1+c_1)\hat{i}+(b_2+c_2)\hat{j}+(b_3+c_3)\hat{k}$

$\text{Now}\ \ \ \ \text{L.H.S}=\vec{a}\times\big(\vec{b}+\vec{c}\big)= \begin{vmatrix}\hat{i} & \hat{j} & \hat{k}\\ a_1& a_2 & a_3 \\ b_1+c_1& b_2+c_2 & b_3+c_3 \end{vmatrix}=\begin{vmatrix}\hat{i} & \hat{j} & \hat{k}\\ a_1& a_2 & a_3 \\ b_1& b_2& b_3\end{vmatrix}+\begin{vmatrix}\hat{i} & \hat{j} & \hat{k}\\ a_1& a_2 & a_3 \\ c_1& c_2& c_3\end{vmatrix}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{[By Property of Determinants]}$

$=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}=\text{R.H.S}.$

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