Originally Posted by
HallsofIvy No, that's perfectly valid- it is a line that passes through (0, 0, 0) when t= 0.
Although I think it is bad practice to use "x" to represent the (x,y,z) coordinates of a point on the line and also use "x" to represent the x coordinate of a point on the plane.
Better would have been $\displaystyle r= \begin{pmatrix}x \\ y \\ z\end{pmatrix}= t\begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix}$.
From that x= 2t, y= t, and z= 3t. Put those into the equation of the plane. If you can solve for t, the line intersects the plane at that point. If you can't solve for t, then the line is parallel to the plane.
Of course, since the equation is linear, the only reason you couldn't solve for t would be that the coefficient of t is 0 which is exactly the same as saying that the dot product of the two vectors Soroban gives is 0- that those two vectors are perpendicular.