determine the constant so that the points is in a plane.
I started to put up two vectors with the points A,B,C:
now i thought that if i choose so that the two vectors makes a plane, i can get the equation for the plane and proof that the point D is in the plane.
How do i choose so the vectors makes a plane?
So, you could, in fact, write the equation of the line, with in the coefficients, then put the coordinates of each point and find what value of is required so that each point satisfies the equation of the plane.
However, simpler is what Plato says- the cross product of two vectors in the plane is perpendicular to the plane and so has 0 dot product with any vector in the plane. That is, you can take the cross product of any two of the vectors, say AC ahd AD and then take the dot product of that with the third, AB. The four points will lie in one plane if and only if that is 0.