## Area The area of a parallelogram is equal to the [[../../02 Areas/Math/Cross Product]] of two of its vectors. ![[../../00 Excalidraw/Cross Product .excalidraw.dark.svg]] ### Finding the area of a triangle Any triangle can be duplicated into a parallelogram. ![[../../00 Excalidraw/Parallelogram _0.excalidraw.dark.svg]] #### Finding in $\R^2$ > [!example] > Find the area of $\triangle PQR$. > $\triangle PQR \in \R^{2}$ > $\huge\begin{align*} > P &= (0, 2) \\ > Q &= (3, 1) \\ > R &= (2, 4) \\\\ > > \alpha &\in \R^{3}\\ > \alpha &: (s, t, 0) \\ > \end{align*}$ > > $\huge\begin{align*} > \vec u &= \proj{\alpha}{\vector{PQ}} = \mat{3\\-1\\0} \\ > \vec v &= \proj{\alpha}{\vector{PR}} =\mat{2\\2\\0} \\ > > \end{align*}$ > > $\huge\begin{align*} > \vec u \times \vec v &= \mat{0\\0\\u_{x}v_{y} - u_{y}v_{x}} \\ > \norm{\vec u \times \vec v} &= \sqrt{ (u_{x}v_y-u_yv_x)^2 }\\ > &=\left|{u_{x}v_y-u_yv_x} \right|\\ > &= 8 > \end{align*}$ > [!abstract] > In general, when finding the area of a triangle or parallelogram in $\R^2$, you can project the vectors on to any plane in $\R^3$ and compute the cross product.