Hello again, Jenny!
2) If $\displaystyle A$ is a plane containing the points $\displaystyle P_1$ and $\displaystyle P_2$
and $\displaystyle P_3$ is any other point in 3space, and: .$\displaystyle \overrightarrow{P_1P_2}\cdot\overrightarrow{P_1P_3 } \:=\:0$,
then $\displaystyle \vec{P_1P_3}$ must be a normal vector for the plane.
The answer is FALSE.
$\displaystyle P_1$ and $\displaystyle P_2$ are on the plane.
$\displaystyle P_3$ is any point not on the plane. Code:
*P3

**
/  /
/  /
/ *     * /
/ P1 P2 /
/ A /
**
Since $\displaystyle \overrightarrow{P_1P_2}\cdot\overrightarrow{P_1P_3 } \:=\:0$, then $\displaystyle \overrightarrow{P_1P_3} \perp \overrightarrow{P_1P_2}$
But $\displaystyle \overrightarrow{P_1P_3}$ is not necessarily perpendicular to the plane.