[Math] Find the vector passing through a given point which is orthogonal to a given triangle in space

Question

I'm given this problem where I have 3 points in space $A(3, -1, 2)$, $B(-2,1,2)$ and $C(2, 0, 5)$.

I need to find the vector passing through point $A$ that is perpendicular to the triangle made by $ABC$.

I'm fairly certain that to find the vector perpendicular to the triangle $ABC$ I'd just need to find the cross product of $AB$ and $AC$. However I'm stuck on how to put this into a vector that passes through $A$.

Answer 1

You would have that $AB \times AC$ is perpendicular to the triangle. Then, the line passing through AB with that direction vector is $OA + t (AB \times AC)$ where $O$ is the origin.