Algebra question about Triangle Interiors I was reading about Triangle Interiors on Wolfram Alpha:
http://mathworld.wolfram.com/TriangleInterior.html
and they have a simple equation:
$$\mathbf{v} = \mathbf{v}_0 + a\mathbf{v}_1 + b\mathbf{v}_2,$$
and then they solve for $a$ and $b$, but I wasn't sure about the steps of how they went about doing that.
My algebra is rusty...
 A: The page defines a binary function $\times$ which maps vectors in $\mathbb{R}^2$ to real numbers, $\mathbb{R}$.  This function is given as $\mathbf{v} \times \mathbf{w} = v_xw_y - w_xv_y$.  It can be shown that it is a bilinear function, since
$$(\mathbf{a} + \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \times \mathbf{c}) + (\mathbf{b} \times \mathbf{c})$$
$$\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c})$$
$$(\lambda\mathbf{a})\times\mathbf{b}=\mathbf{a}\times(\lambda\mathbf{b})=\lambda(\mathbf{a}\times\mathbf{b})$$
$\forall\mathbf{a},\mathbf{b},\mathbf{c}\in\mathbb{R^2},\forall\lambda\in\mathbb{R}$ (to see this, set $\mathbf{a}=(a_x,a_y),$ $\mathbf{b}=(b_x,b_y),$ and $\mathbf{c}=(c_x,c_y),$ then substitute into the above).  Using the above equations
$$\mathbf{v}\times\mathbf{v_1}=(\mathbf{v_0} + a\mathbf{v_1} + b\mathbf{v_2})\times\mathbf{v_1}$$
and
$$\mathbf{v}\times\mathbf{v_2}=(\mathbf{v_0} + a\mathbf{v_1} + b\mathbf{v_2})\times\mathbf{v_2}$$
can be manipulated to obtain the desired results, so long as $\mathbf{v_1}\times\mathbf{v_2}\ne0$.
