Given two vectors vec{A} = 3hat{i} + hat{j} a | vedclass

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(C) The area of a parallelogram formed by two adjacent vectors $\vec{A}$ and $\vec{B}$ is given by the magnitude of their cross product: $	ext{Area}

Vedclass · Accepted Answer

$7$

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(C) The area of a parallelogram formed by two adjacent vectors $\vec{A}$ and $\vec{B}$ is given by the magnitude of their cross product: $	ext{Area} = |\vec{A} 	imes \vec{B}|$.First,calculate the cross product $\vec{A} 	imes \vec{B}$:$\vec{A} 	imes \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & 1 & 0 \ 0 & 1 & 2 \end{vmatrix}$$= \hat{i}(1 	imes 2 - 0 	imes 1) - \hat{j}(3 	imes 2 - 0 	imes 0) + \hat{k}(3 	imes 1 - 1 	imes 0)$$= \hat{i}(2) - \hat{j}(6) + \hat{k}(3) = 2\hat{i} - 6\hat{j} + 3\hat{k}$.Now,calculate the magnitude of this vector:$|\vec{A} 	imes \vec{B}| = \sqrt{(2)^2 + (-6)^2 + (3)^2}$$= \sqrt{4 + 36 + 9} = \sqrt{49} = 7$.Thus,the area of the parallelogram is $7$ square units.

Given two vectors $\vec{A} = 3\hat{i} + \hat{j}$ and $\vec{B} = \hat{j} + 2\hat{k}$. If these two vectors represent the adjacent sides of a parallelogram,find the area of the parallelogram.

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