A line through (1, –1, 2) with direction ratios <3, 2, 2> meets the plane x + 2y + 3z = 18. What is the point of intersection of line and plane?

Parametrize the line as (1 + 3t, –1 + 2t, 2 + 2t). Substitute into x + 2y + 3z = 18: (1 + 3t) + 2(–1 + 2t) + 3(2 + 2t) = 18 ⟹ 1 + 3t – 2 + 4t + 6 + 6t = 18 ⟹ 5 + 13t = 18 ⟹ t = 1. So the point is (1 + 3, –1 + 2, 2 + 2) = (4, 1, 4).