Q.Find the area of the triangle with vertices $(1,1,0)$, $(1,0,1)$ and $(0,1,1)$.

Let $A=(1,1,0),B=(1,0,1),C=(0,1,1)$. Form two side vectors from $A$:

Their cross product is

$\overrightarrow{AB}\times\overrightarrow{AC}=\begin{vmatrix}\hat i&\hat j&\hat k\\ 0&-1&1\\ -1&0&1\end{vmatrix} =\hat i(-1\cdot1-1\cdot0)-\hat j(0\cdot1-1\cdot(-1))+\hat k(0\cdot0-(-1)(-1))=-\hat i-\hat j-\hat k.$ …

Form two side vectors from a common vertex, compute their cross product via the determinant formula, then …

Forgetting the factor of 1/2 (which gives the parallelogram's area instead of the triangle's), or a sign/ar …