How to know if a straight line and a plane are parallel?

Question

Given the points $A=(2,5,1)$, $B=(3,0,3)$, $C=(1,2,4)$ and $D=(1,-3,0)$. How can I know if the straight line $r$ passing between $A$ and $B$ and the plane passing between $A$, $C$ and $D$ are parallel?

Answer 1

Well, they are parallel iff $B$ is in a plane $ACD$. So, you must check whether you can express vector $\vec{AB}$ with vectors $\vec{AC}$ and $\vec{AD}$:

$$\vec{AB}= m\vec{AC}+n\vec{AD}$$

for some real $m,n$

Answer 2

you must compute the direction vector of the straight line and a vector which is perpendicular to the plane, the dot product of both should be zero