Answer
(a) Find two vectors between the three points $P,Q$ and $R$; for example, find $\vec{PQ}$ and $\vec{QR}$ and see if they are parallel; that is, scalar multiples of each other. If they are parallel, then the points are on the same line; otherwise, they are not.
(b) Take three points such as $P,Q,R$ to find two vectors ${PQ}$ and ${QR}$.
Take the cross product of ${PQ} \times {QR}$ to find the normal vector $ \lt a,b,c \gt$
Let $P,Q,R$ as $(x_0,y_0,_0)$ be a point on the plane equation: $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$
Three points always lie in a single plane, so we only need to test the 4th point $S$ in the resulting plane equation to see the numbers work. If they do then all four points lie in the same plane.
Work Step by Step
(a) Find two vectors between the three points $P,Q$ and $R$; for example, find $\vec{PQ}$ and $\vec{QR}$ and see if they are parallel; that is, scalar multiples of each other. If they are parallel, then the points are on the same line; otherwise, they are not.
(b) Take three points such as $P,Q,R$ to find two vectors ${PQ}$ and ${QR}$.
Take the cross product of ${PQ} \times {QR}$ to find the normal vector $ \lt a,b,c \gt$
Let $P,Q,R$ as $(x_0,y_0,_0)$ be a point on the plane equation: $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$
Three points always lie in a single plane, so we only need to test the 4th point $S$ in the resulting plane equation to see the numbers work. If they do then all four points lie in the same plane.
