Solutions aren't given, so I kindly ask for a recheck.Vectors $\displaystyle a, b, c$ ($\displaystyle \vec{i}=i$) make uptetrahedronwith volume $\displaystyle V=3$. What's the volume of atetrahedronmade up of vectors $\displaystyle a\times b, b\times c, a\times c$?

Setup: $\displaystyle V=\frac{1}{6}(a,b,c)=3\Rightarrow a\cdot(b\times c)=18\Rightarrow b\times c=\frac{18}{a}$

Attempt:

$\displaystyle (a\times b, b\times c, a\times c)=((a\times b)\times(a\times c))\cdot b\times c=(-\langle b, a\times c\rangle\cdot a)\cdot b\times c=(-\langle b, a\times c\rangle\cdot a)\frac{18}{a}=$

$\displaystyle =-18\langle b, a\times c \rangle=-18(b\cdot (a\times c))=-18(b,a,c)=18(a,b,c)=54$

Isn't the volume of the secondtetrahedrontoo large?