Write a vector an parametric equation for the plane which contains the point P(9, 1, -2) and is parallel to [x, y, z] = (4, 1, 8) + [1, -1, 1) and [x, y, z] = (-5, 0, 10) + t(-6, 2, 5).

Answer 1

slightly puzzled by your notation.
Looks like you are using [....] to represent vector directions, but in
[x, y, z] = (4, 1, 8) + [1, -1, 1) , there is confusion and [1,-1,1] lacks a parameter.
So assuming the two lines have direction [1,-1,1] and [-6,2,5]
We need a normal to both of those, which is the cross-product
one such normal is [7,11,4] , (I assume you know how find that normal)

So the equation of the plane is 7x + 11y + 4z = k
with (9,1,-2) on it, so ....
63 + 11 - 8 = k = 66

7x + 11y + 4z = 66

Answer 2

If you want a parametric equation for the plane, it is very simply:

[x,y,z] = (9, 1, -2) + s[1, -1, 1] + t[-6, 2, 5]

Answer 3

Isn't [x,y,z] = (9, 1, -2) + s[1, -1, 1] + t[-6, 2, 5] the vector equation not the parametric equation?

Answer 4

Surely you are not going to tell me that you don't know how to change the vector equation to parametric form.

here is the x,
x = 9 + s - 6t
how about:
y = ...
z = ...