Question

The area of a parallelogram formed by the vectors Ā = i +2j hat +3k hat and B =3î - 2i+k hat
adjacent sides is?
Answer given - 8√3 units.
solve this asap
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Answer 1

❍ We are provided with the component form of $\overrightarrow{A}$ and $\overrightarrow{B}$

❍ We are asked to find the area of the parallelogram so formed.

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❍ We know, that this can be calculated by:-

$\leadsto \underline{ \boxed{ \pink{{ \bf{area = | \overrightarrow{A} \times \overrightarrow{B} | }}}}} \bigstar$

Let's calculate A vector cross B vector first ...

$\begin {array}{ |c c c | } \sf\hat{i} & \sf \hat{j}& \sf \hat{k} \\ \\ \sf 1 & \sf 2& \sf 3 \\ \\ 3& - 2&1\end{array}$

$\rightarrow \sf \overrightarrow{A} \times \overrightarrow{B} = (2 + 6) \hat{i} - (1 - 9) \hat{j} + ( - 2 - 6) \hat{k}$

$\rightarrow \sf \overrightarrow{A} \times \overrightarrow{B} = 8 \hat{i} + 8 \hat{j}- 8 \hat{k}$

❍ Now, we are going to calculate the magnitude of A vector cross B vector which will be the area of the parallelogram.

❍ we know that cross product of two vectors is always a vector.

❍ The magnitude of this vector can be determined by the following formula:-

$\leadsto \underline{ \boxed{ \pink{{ \bf{| \overrightarrow{A}|=\sqrt{{A_x^{2} } +A_y^{2} +A_z^{2} } }}}}} \bigstar$

❍ Here, Ax , Ay and Az are components of A vector along x, y and z axis respectively.

$\dag\underline{\frak{Applying\: same\: formula\: for\: our\:equation:- }}$

$: \implies \sf | \overrightarrow{A} \times \overrightarrow{B}| = \sqrt{ {8}^{2} + { - 8}^{2} + {8}^{2} }$

$: \implies \boxed {\pink{{\sf | \overrightarrow{A} \times \overrightarrow{B}| = 8 \sqrt{3} \: sq \: units} }}\bigstar$

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I hope it's helpful :D