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  • The area of the parallelogram formed from the vectors <img alt="\vec{A}=\hat{l}-2\hat{j}+3\hat{k}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Cvec%7BA%7D%3D%5Chat%7Bl%7D-2%5Chat%7Bj%7Da

The area of the parallelogram formed from the vectors \vec{A}=\hat{l}-2\hat{j}+3\hat{k}  and  \vec{B}=3\hat{l}-2\hat{j}+\hat{k}   as adjacent side is

Option: 1

8\sqrt{3}\; \; units\; \;


Option: 2

\; \; 64\; \; units\; \; \;


Option: 3

\; 32\; \; units\; \;


Option: 4

\; 4\sqrt{6}\; \; units


Answers (1)

best_answer

As we learned

Vector or cross product -

Vector or cross product of two vector \vec{A} & \vec{B} written asA\times B is a single vector whose magnitude is equal to product of magnitude of \vec{A} & \vec{B} and the sine of smaller angle\Theta  between them.

\vec A\times \vec B= A\, B\sin \Theta

- wherein

Figure \shows a representation of vector or cross product of vectors.

 

shows representation of vector or cross product of vectors

 

 Area of parallelogram  is \left | \vec{A}\times \vec{B} \right |

\\*\vec{A}\times \vec{B}=\begin{vmatrix} \hat{l} & \hat{j} &\hat{k} \\ 1 &-2 &3 \\ 3&-2 & 1 \end{vmatrix}\\*\\*=\hat{l}(-2+6)-\hat{j}(1-9)+\hat{k} (-2+6)\\*\\*=4\: \hat{l}+8\hat{j}+4\hat{k}\\*\\*\therefore Area\; \; =\left | \vec{A}\times \vec{B} \right |=\sqrt{4^{2}+8^{2}+4^{2}}=\sqrt{96}=4\sqrt{6}\; units

 

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Rishi

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