I'm used to a slightly different form to write vectors so you have to change my results into the form you are used to. Sorry.

A(1, 1, 2)

B(1, 2, 3)

C(3, 0, 1)

to #1. Probably you know that the absolute value of the cross-product of 2 non-collinear vectors equals the area of the parallelogram which is spanned by the vectors. Thus the area of the triangle ABC is:

......... Therefore you have:

to #3:

The cross-product yields the normal vector of the plane. Therefore the equation of the plane is:

to #4:

The line L has the equation:

with

x=0

y=2t

z=-2t

Plug in these values into the equation of the palne and solve for t:

Plug in this value into the equation of the line to get the coordinates of D:

to #5:

Rewrite the equation of the plane so the normal vector becomes a unit vector:

The equation of the plane becomes now: