The area of the parallelogram spanned by two vectors is the magnitude of their cross product. The area of a triangle is half the area of a parallelogram. The parallelogram form by A, B, and C is spanned by
and
which in 1(a), we already determined were
![{\displaystyle {\begin{aligned}{\vec {AB}}&=[-2,2,1]\\{\vec {AC}}&=[-1,1,1].\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d22b4eeca77a415b2bbe3bb956e20af902f47e8b)
We need the cross product of this, but as we saw in 1(a), this is just the normal vector to the plane
and therefore
. Therefore the area of the triangle is
![{\displaystyle {\textrm {area}}={\frac {1}{2}}||{\vec {AB}}\times {\vec {AC}}||={\frac {1}{2}}||\mathbf {m} ||={\frac {1}{2}}{\sqrt {1^{2}+1^{2}+0^{2}}}={\frac {1}{2}}{\sqrt {2}}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/df700bc1d90bc0edae776eb0789ee52047fbbcb3)