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સદિશ બીજગણિત — ગણિત ધોરણ 12 સાયન્સ — Question

Gujarat Boardગુજરાતી માધ્યમધોરણ 12 સાયન્સગણિતસદિશ બીજગણિત1 Mark
MCQ
વિષમતલીય સદિશો $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ માટે $(\overrightarrow{a} \times \overrightarrow{b}) +(\overrightarrow{b} \times \overrightarrow{c})+(\overrightarrow{c} \times \overrightarrow{a}) = \ .....$
  • A
    $(\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{c}-\overrightarrow{b})$
  • ✓
    $(\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{a}-\overrightarrow{c})$
  • C
    $(\overrightarrow{a}-\overrightarrow{c}) \times (\overrightarrow{c}-\overrightarrow{b})$
  • D
    $\frac{1}{2} \left[(\overrightarrow{a}\times (\overrightarrow{b}-\overrightarrow{c})-\overrightarrow{b}\times (\overrightarrow{c}-\overrightarrow{a})+\overrightarrow{c}(\overrightarrow{a}-\overrightarrow{b})\right]$
✓

Answer

Correct option: B.
$(\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{a}-\overrightarrow{c})$
$(\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{c}-\overrightarrow{b})=\overrightarrow{a} \times \overrightarrow{c}-\overrightarrow{a} \times \overrightarrow{b}-\overrightarrow{b} \times \overrightarrow{c}$ $\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{c} \times \overrightarrow{a}$
$\therefore (A)$ સત્ય નથી.
$($ નોંધ : $\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{c} \times \overrightarrow{a}$ શૂન્ય સદિશ નથી. ખાતરી કરો.$)$
હવે, $(\overrightarrow{a}-\overrightarrow{b}) \times (\overrightarrow{a}-\overrightarrow{c})=-\overrightarrow{a} \times \overrightarrow{c}-\overrightarrow{b} \times \overrightarrow{a}+\overrightarrow{b} \times \overrightarrow{c}=\overrightarrow{c} \times \overrightarrow{a}+\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}$
$\therefore (B)$ સત્ય છે.
$(\overrightarrow{a}-\overrightarrow{c}) \times (\overrightarrow{c}-\overrightarrow{b})=-\overrightarrow{a} \times \overrightarrow{c}-\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{c} \times \overrightarrow{b}=-(\overrightarrow{c} \times \overrightarrow{a}+\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c})$
$\therefore (C)$ સત્ય નથી.
વળી, $\frac{1}{2}[(\overrightarrow{a} \times (\overrightarrow{b}-\overrightarrow{c})+\overrightarrow{b} \times (\overrightarrow{c}-\overrightarrow{b})+\overrightarrow{c} \times (\overrightarrow{a}-\overrightarrow{b}))]$
$=\frac{1}{2}[\overrightarrow{a} \times \overrightarrow{b}-\overrightarrow{a} \times \overrightarrow{c}+\overrightarrow{b} \times \overrightarrow{c}-\overrightarrow{b} \times \overrightarrow{a}+\overrightarrow{c} \times \overrightarrow{a}-\overrightarrow{c} \times \overrightarrow{b}]$
$=\frac{1}{2}[\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{c} \times \overrightarrow{a}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{c} \times \overrightarrow{a}+\overrightarrow{b} \times \overrightarrow{c}]$
$=\frac{1}{2}(2)(\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{c} \times \overrightarrow{a})$
$=\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{c} \times \overrightarrow{a}$
$\therefore (D)$ સત્ય છે.

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