Question

Determine if the point (4,-2,3) lies in the plane with vector equation (x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1, 2).

Answer 1

we need
2+4s-3t = 4
0-2s-t = -2
-1+s+2t = 3

Now just see whether there is a solution which fits all three equations.