Let $ u$, $v$ and $ w$ be vectors in R3 such that $u* v = (1, 0,-1)$ and $w *u = (-1,0, 0).$ Find the area of the parallelogram defined by $u$ and $(v+w)$.
So my thought was: $u*(v+w) = u*v + u*w$. I know that the parallelogram defined by $u∗v$ has area $\sqrt2$ but I can't find the area of the other one. Any help on this?
You have $u*v$ and $w*u$. Remember that $u*w=-w*u$.
Now, calculate the vector:
$$u*v-w*u=(1,0,-1)-(-1,0,0)=(2,0,-1)$$
The area will bee then
$$|(2,0,-1)|=\sqrt{5}$$