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Question

# Find the areas of the parallelograms whose vertices are given. A(1, 0, -1), B(1, 7, 2), C(2, 4, -1), D(0, 3, 2)

Solution

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$\mathbf{u}=\mathbf{c}-\mathbf{a}= \langle2,4,-1\rangle-\langle1,0,-1 \rangle=\langle1,4,0\rangle$

$\mathbf{v}=\mathbf{d}-\mathbf{a}= \langle0,3,2\rangle-\langle1,0,-1 \rangle=\langle-1,3,3\rangle$

$\mathbf{u}\times\mathbf{v}= \left| \begin{matrix} \textbf{i} & \textbf{j} & \textbf{k} \\ 1 & 4 & 0 \\ -1 & 3 & 3 \end{matrix}\right|\\\\ =\textbf{i}\left|\begin{matrix} 4 & 0\\ 3 & 3 \end{matrix}\right|-\textbf{j} \left|\begin{matrix} 1 & 0\\ -1 & 3 \end{matrix}\right|+\textbf{k} \left|\begin{matrix} 1 & 4\\ -1 & 3 \end{matrix}\right|$

$=(12-0)\mathbf{i}-(3-0)\mathbf{j}+(3+4)\mathbf{k}$

$=\langle12,-3,7\rangle$

$A=\left|\mathbf{u}\times\mathbf{v} \right|$

$=\sqrt{(12)^2+(-3)^2+(7)^2}$

$=\sqrt{202}$

Compute two vector representations $\textbf{u}$ and $\textbf{v}$ from the set of points with respect to the arbitrary point $A$. Then compute their cross product and its resulting magnitude in order to determine the area of the parallelogram..

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