someone could show me the error in the cross products?
For $U=x\hat{i}+y\hat{j}+z\hat{k}$, $V=x'\hat{i}+y'\hat{j}+z'\hat{k}$ and $((.))$=modulus, we have
$$U \times V=((U))((V))sin(U,V).n = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ x' & y' & z' \end{vmatrix} $$
(1) $U=2\hat{i}$, $V=160(-\hat{j})$ and $sin(2\hat{i},-160\hat{j})=1$ $$U\times V=2\hat{i} \times (-160)\hat{j}=((2\hat{i}))((-160\hat{j})).(\hat{i} \times \hat{j})=2.160.\hat{k}=320\hat{k}$$
(2) $2\hat{i} \times (-160)\hat{j}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ 2 & 0 & 0\\ 0 & -160 & 0 \end{vmatrix}=-320\hat{k}$
They just don't match..
Have a nice day.
The normal vector $\vec n$ has to be not just perpendicular, but in the direction specified by the right-hand rule. So it is $-\vec k$, not $\vec k$