Find the coordinates of the point of intersection between the line L:(-i+j-5k)+v(i+j+2k) and the plane π: r.(i+2j+3k)=4.

By inspection we can tell that the line and the plane are not parallel. Since the dot product between the direction vector of the line with the perpendicular vector of the plane was not equal to 0 (i.e. b.n≠0). If they were to be parallel you should check that the line is not in the plane, by testing point A of the line in the equation of the plane.
Therefore we can find the point of intersection by doing the dot product of a point on the line with the vector (i+2j+3k) and setting it equal to 4.After some rearranging we find that v=2.we can sub this into the equation of the line to find the vector point of intersection.Finally write as coordinates (1,2,3) and check point lies in the line and the plane just to be sure.