How can one calculate the vector normal to the plane that is determined by given points?
For example, given three points $P_1(5,0,0)$, $P_2(0,0,5)$ and $P_3(10,0,5)$, calculate the vector normal to the plane containing these three points.
The compute the normal is by vector product. $ a = \left(\begin{matrix} x_2-x_1\\y_2-y_1\\z_2-z_1 \end{matrix} \right) \qquad b = \left(\begin{matrix} x_3-x_1\\y_3-y_1\\z_3-z_1 \end{matrix} \right) $ therefore $a = -5i+5k$, and $b=5i+5k$ $ a\times b = \left|\begin{matrix} i & j & k \\ -5 & 0 & 5 \\ 5 & 0 & -5 \\ \end{matrix}\right| $
The questions are:
1) Are A and B is given, that means no matter which three point i use, the $a$ and $b$ is still using this to calucate?
2) How to get $a$ , $b$ and $a\times b$ ? Thank you