Given that a = i + $2$j + $3$k, b = $2$i - j and c = j - $2$k, verify that a$\cdot$b $\times$ c = b$\cdot$c $\times$ a = c$\cdot$a $\times$ b.
So I took
a$\cdot$b $\times$ c = $\left(\begin{matrix}1\\2\\3\\\end{matrix}\right)\cdot \left(\begin{matrix}2\\-1\\0\\\end{matrix}\right)\times\left(\begin{matrix}0\\1\\-2\\\end{matrix}\right)=0$
However, b$\cdot$c $\times$ a doesn’t seem to yield the same value as a$\cdot$b $\times$ c .
b$\cdot$c $\times$ a $=\left(\begin{matrix}2\\-1\\0\\\end{matrix}\right)\cdot\left(\begin{matrix}0\\1\\-2\\\end{matrix}\right)\times\left(\begin{matrix}1\\2\\3\\\end{matrix}\right)$= $-1 \left(\begin{matrix}1\\2\\3\\\end{matrix}\right)$
Did I multiply anything wrong?
You should get a final result of $16$ for all the products. Your result of a vector for $\mathbf b\cdot\mathbf c×\mathbf a$ is wrong – for expressions of this type the cross product must be performed first.