Solution: The parallel line through the point has the vector equation L = < 0, 1, 2 > + t < a, b, c >, and we have to determine a, b, and c.
( 1 ) L is parallel to the plane x + y + z = 2
L is perpendicular to n = < 1, 1, 1 >, the normal to the plane.
L and n are perpendicular
Thus a + b + c = 0.
( 2 ) L and the line < 1, 1, 0 > + t < 1, -1, 2 > are perpendicular
Thus a - b + 2c = 0
We have two equations with three variables:
a - b = -2c
Solving for a and b in terms of c :
2b = c b = c/2
Thus the vector equation of the line is : L = < 0, 1, 2 > + t < 3, -1, -2 > and the parametric equations are: