Question

pls help me as fast as u can​

Answer 1

Answer:

Let the two vectors be :

$\implies \sf \overrightarrow{A} =\widehat{i} + 2\widehat{j} + 3\widehat{k}$

$\implies \sf \overrightarrow{B} =3\widehat{i} - 2\widehat{j} + \widehat{k}$

Area of parallelogram is calculated by the given below formula:

$\implies \sf Area \: of \: parallelogram = |\overrightarrow{A} \times \overrightarrow{B}|$

Calculation:

$\longrightarrow\: \displaystyle\sf \overrightarrow{A} \times \overrightarrow{B} = \left(</p><p>\begin{array}{ccc} \sf\widehat{i} & \sf \widehat{j} &\sf\widehat{k} \\\\\tt1& \tt2& \tt 3 \\\\ \tt3 & \tt (- 2) & \tt1</p><p>\end{array}</p><p>\right)$

$\longrightarrow\: \displaystyle\sf \overrightarrow{A} \times\overrightarrow{B} = \widehat{i}\Bigg\{2 \times 1 - 3 \times ( - 2)\Bigg\} - \widehat{j}\Bigg\{1 \times 1 - 3 \times 3\Bigg\} + \widehat{k}\Bigg\{1 \times ( - 2) - 2 \times 3\Bigg\}$

$\longrightarrow\: \displaystyle\sf \overrightarrow{A} \times \overrightarrow{B} = \widehat{i}\Bigg\{2 + 6\Bigg\} - \widehat{j}\Bigg\{1 - 9\Bigg\} + \widehat{k}\Bigg\{- 2 - 6\Bigg\}$

$\longrightarrow\: \displaystyle\sf \overrightarrow{A} \times \overrightarrow{B} =8 \widehat{i} + 8\widehat{j} - 8\widehat{k}$

$\longrightarrow\: \displaystyle\sf | \overrightarrow{A} \times \overrightarrow{B} | = \sqrt{ {(8)}^{2} + (8)^{2} + ( - 8 {)}^{2} }$

$\longrightarrow\: \displaystyle\sf | \overrightarrow{A} \times \overrightarrow{B} | = \sqrt{ 64 + 64 + 64 }$

$\longrightarrow\: \displaystyle\sf | \overrightarrow{A} \times \overrightarrow{B} | =\sqrt{ 192 }$

$\longrightarrow\: \displaystyle\sf | \overrightarrow{A} \times \overrightarrow{B} | =\sqrt{ 24 \times 8}$

$\longrightarrow\: \displaystyle\sf | \overrightarrow{A} \times \overrightarrow{B} | =\sqrt{ 8 \times 3 \times 8}$

$\longrightarrow\: \displaystyle \underline{ \boxed{ \red{\bf | \overrightarrow{A} \times \overrightarrow{B} | =8 \sqrt{ 3 }}}}$

Hence, Option C) is the correct answer.