Q.Find the area of parallelogram whose diagonals are determined by the vectors $\bar a=3\hat i-\hat j-2\hat k$, and $\bar b=-\hat i+3\hat j-3\hat k$.

If a parallelogram has diagonal vectors $\bar p,\bar q$ from a common vertex, its area is $\tfrac12|\bar p\times\bar q|$ (this follows since the diagonals of a parallelogram with sides $\bar u,\bar v$ are $\bar\nu+\bar v$ and $\bar u-\bar v$, and $(\bar u+\bar v)\times(\bar u-\bar v)=-2(\bar u\times\bar v)$, so $|\bar p\times\bar q|=2|\bar u\times\bar v|=2\times$area).

Here $\bar a=3\hat i-\hat j-2\hat k,\ \bar b=-\hat i+3\hat j-3\hat k$: …

Recall that for a parallelogram given by its two diagonal vectors, the area is half the magnitude of their cross product; …

Using the diagonal-vectors' cross product directly as the area without halving (confusing the diagonal formula with th …