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ત્રિપરિમાણીય ભૂમિતિ — ગણિત ધોરણ 12 સાયન્સ — Question

Gujarat Boardગુજરાતી માધ્યમધોરણ 12 સાયન્સગણિતત્રિપરિમાણીય ભૂમિતિ1 Mark
MCQ
સદિશો $\left(\left(\hat{i}\times\overrightarrow{a}\right)\cdot\overrightarrow{b}\right)\hat{i}+\left(\left(\hat{j}\times\overrightarrow{a}\right)\cdot\bar{b}\right)\hat{j}+\left(\left(\hat{k}\times\overrightarrow{a}\right)\cdot\overrightarrow{b}\right)\hat{k}=\ .......$
  • A
    $\overrightarrow{b}\times\overrightarrow{a}$
  • B
    $\overrightarrow{a}$
  • ✓
    $\overrightarrow{a}\times\overrightarrow{b}$
  • D
    $\overrightarrow{b}$
✓

Answer

Correct option: C.
$\overrightarrow{a}\times\overrightarrow{b}$
ધારો કે $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ અને $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$
હવે $(\hat{i}\times\overrightarrow{a})\cdot\overrightarrow{b}=\ \ [\hat{i} \ \ \overrightarrow{a}\ \ \ \overrightarrow{b}]$
$=\begin{vmatrix}1 & 0 & 0 \\a_1 & a_2 & a_3 \\b_1 & b_2 & b_3\end{vmatrix}=a_2b_3-a_3b_2$
તે જ પ્રમાણે $(\hat{j}\times\overrightarrow{a})\cdot\overrightarrow{b}=a_3b_1-a_1b_3$ અને $(\hat{k}\times\overrightarrow{a})\cdot\overrightarrow{b}=a_1b_2-a_2b_1$
$\therefore((\hat{i}\times\overrightarrow{a})\cdot\overrightarrow{b})\hat{i}+((\hat{j}\times\overrightarrow{a})\cdot\overrightarrow{b})\hat{j}+((\hat{k}\times\overrightarrow{a})\cdot\overrightarrow{b})\hat{k}$
$=(a_2b_3-a_3b_2)\hat{i}+(a_3b_1-a_1b_3)\hat{j}+(a_1b_2-a_2b_1)\hat{k}$
$=\overrightarrow{a}\times\overrightarrow{b}$
બીજી રીત:
$\left(\left(\hat{i}\times\hat{a}\right)\cdot\hat{b}\right)\hat{i}+\left(\left(\hat{j}\times\hat{a}\right)\cdot\hat{b}\right)\hat{j}+\left(\left(\hat{k}\times\hat{a}\right)\cdot\hat{b}\right)\hat{k}$
$=\left((\hat{i}\cdot(\hat{a}\times\hat{b})\right)\hat{i}+\left((\hat{j}\cdot(\hat{a}\times\hat{b})\right)\hat{j}+\left((\hat{k}\cdot(\hat{a}\times\hat{b})\right)\hat{k}$
$=\overrightarrow{a}\times\overrightarrow{b}$

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