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# Area of Parallelogram with two adjacent 2d vectors.

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Find the area of the parallelogram with u and v as adjacent edges.

u = 5i -2j

v = 6i -2j

Jun 11, 2020

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The area is equal to $$\left|\mathbf{u} \times \mathbf{v}\right|$$.

Notice that $$\mathbf{u}\times \mathbf{v} = \det\begin{bmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\5&-2&0\\6&-2&0\end{bmatrix} = \det\begin{bmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\5&-2&0\\1&0&0\end{bmatrix} = 2\mathbf{k}$$

Therefore $$\left|\mathbf{u} \times \mathbf{v}\right| = |2\mathbf{k}| = 2$$

Therefore the area of the required parallelogram is 2 sq. units.

Jun 11, 2020