Question

how to find the cartesian eqn of plane passing through A(0,0,0) and B(3,-1,2) and parellel to line (x-4)/1 = (y+3)/-4 = (z+1)/7

basically how to find a plane parallel to a given line??

Answer 1

We need two direction vectors in the plane, one is vector AB or (3, -1, 2)

the other is the direction of the line parallel to the plane which is ((1, -4, 7)

a normal to the plane would be the cross - product of the two vectors above

Assuming you know how to find that normal, I got
(1 , -19, -11)

So the plane has equation,
x - 19y - 11z = c
but (0,0,0) lies on it,

so

x - 19y -11z = 0